System and method for communication using orbital angular momentum with multiple layer overlay modulation

ABSTRACT

A communications system includes a transmitter for transmitting an optical signal including a plurality of data streams over an optical communications link. The transmitter includes first signal processing circuitry for processing each of the plurality of input data streams to generate a parallel pair of data streams including an in-phase stream (I) and a quadrature-phase stream (Q) for each of the plurality of input data streams. The first signal processing circuitry modulates a first and second parallel pair of data streams with a selected one of at least three mutually orthogonal functions at a first and second signal widths, respectively, to generate a plurality of first data sub-layers and a plurality of second data sub-layers and generates a plurality of composite data stream by overlaying the first data subs-layers with the second data sub-layers at a preconfigured overlay offset. Second signal processing circuitry places the plurality of composite data streams on a wavelength of the optical communications link. Each of the plurality of composite data streams having a different orbital angular momentum associated therewith to enable transmission of each of the plurality of composite data streams on the wavelength at a same time.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims benefit of U.S. Provisional Application No.61/975,142, filed Apr. 4, 2014, entitled SYSTEM AND METHOD FORCOMMUNICATION USING ORBITAL ANGULAR MOMENTUM WITH MODULATION (Atty. Dkt.No. NXGN-32131), the specification of which is incorporated by referenceherein in its entirety.

TECHNICAL FIELD

The following disclosure relates to systems and methods for increasingcommunication bandwidth, and more particularly to increasingcommunications bandwidth using a combination of the application oforbital angular momentum to various signals, and the modulation ofsignals using a multiple layer overlay modulation scheme.

BACKGROUND

The use of voice and data networks has greatly increased as the numberof personal computing and communication devices, such as laptopcomputers, mobile telephones, Smartphones, tablets, et cetera, hasgrown. The astronomically increasing number of personal mobilecommunication devices has concurrently increased the amount of databeing transmitted over the networks providing infrastructure for thesemobile communication devices. As these mobile communication devicesbecome more ubiquitous in business and personal lifestyles, theabilities of these networks to support all of the new users and userdevices has been strained. Thus, a major concern of networkinfrastructure providers is the ability to increase their bandwidth inorder to support the greater load of voice and data communications andparticularly video that are occurring. Traditional manners forincreasing the bandwidth in such systems have involved increasing thenumber of channels so that a greater number of communications may betransmitted, or increasing the speed at which information is transmittedover existing channels in order to provide greater throughput levelsover the existing channel resources.

However, while each of these techniques have improved system bandwidths,existing technologies have taken the speed of communications to a levelsuch that drastic additional speed increases are not possible, eventhough bandwidth requirements due to increased usage are continuing togrow exponentially. Additionally, the number of channels assigned forvoice and data communications, while increasing somewhat, have notincreased to a level to completely support the increasing demands of avoice and data intensive use society. Thus, there is a great need forsome manner for increasing the bandwidth throughput within existingvoice and data communication that increases the bandwidth on existingvoice and data channels.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding, reference is now made to thefollowing description taken in conjunction with the accompanyingDrawings in which:

FIG. 1 illustrates various techniques for increasing spectral efficiencywithin a transmitted signal;

FIG. 2 illustrates a particular technique for increasing spectralefficiency within a transmitted signal;

FIG. 3 illustrates a general overview of the manner for providingcommunication bandwidth between various communication protocolinterfaces;

FIG. 4 illustrates the manner for utilizing multiple level overlaymodulation with twisted pair/cable interfaces;

FIG. 5 illustrates a general block diagram for processing a plurality ofdata streams within an optical communication system;

FIG. 6 is a functional block diagram of a system for generating orbitalangular momentum within a communication system;

FIG. 7 is a functional block diagram of the orbital angular momentumsignal processing block of FIG. 6;

FIG. 8 is a functional block diagram illustrating the manner forremoving orbital angular momentum from a received signal including aplurality of data streams;

FIG. 9 illustrates a single wavelength having two quanti-spinpolarizations providing an infinite number of signals having variousorbital angular momentums associated therewith;

FIG. 10A illustrates a plane wave having only variations in the spinangular momentum;

FIG. 10B illustrates a signal having both spin and orbital angularmomentum applied thereto;

FIGS. 11A-11C illustrate various signals having different orbitalangular momentum applied thereto;

FIG. 11D illustrates a propagation of Poynting vectors for various Eigenmodes;

FIG. 11E illustrates a spiral phase plate;

FIG. 12 illustrates a multiple level overlay modulation system;

FIG. 13 illustrates a multiple level overlay demodulator;

FIG. 14 illustrates a multiple level overlay transmitter system;

FIG. 15 illustrates a multiple level overlay receiver system;

FIGS. 16A-16K illustrate representative multiple level overlay signalsand their respective spectral power densities;

FIG. 17 illustrates comparisons of multiple level overlay signals withinthe time and frequency domain;

FIG. 18 illustrates a spectral alignment of multiple level overlaysignals for differing bandwidths of signals;

FIG. 19 illustrates an alternative spectral alignment of multiple leveloverlay signals;

FIG. 20 illustrates power spectral density for various signal layersusing a combined three layer multiple level overlay technique;

FIG. 21 illustrates power spectral density on a log scale for layersusing a combined three layer multiple level overlay modulation;

FIG. 22 illustrates a bandwidth efficiency comparison for square rootraised cosine versus multiple layer overlay for a symbol rate of 1/6;

FIG. 23 illustrates a bandwidth efficiency comparison between squareroot raised cosine and multiple layer overlay for a symbol rate of 1/4;

FIG. 24 illustrates a performance comparison between square root raisedcosine and multiple level overlay using ACLR;

FIG. 25 illustrates a performance comparison between square root raisedcosine and multiple lever overlay using out of band power;

FIG. 26 illustrates a performance comparison between square root raisedcosine and multiple lever overlay using band edge PSD;

FIG. 27 is a block diagram of a transmitter subsystem for use withmultiple level overlay;

FIG. 28 is a block diagram of a receiver subsystem using multiple leveloverlay;

FIG. 29 illustrates an equivalent discreet time orthogonal channel ofmodified multiple level overlay;

FIG. 30 illustrates the PSDs of multiple layer overlay, modifiedmultiple layer overlay and square root raised cosine;

FIG. 31 illustrates a bandwidth comparison based on −40 dBc out of bandpower bandwidth between multiple layer overlay and square root raisedcosine;

FIG. 32 illustrates equivalent discrete time parallel orthogonalchannels of modified multiple layer overlay;

FIG. 33 illustrates the channel power gain of the parallel orthogonalchannels of modified multiple layer overlay with three layers and T=3;

FIG. 34 illustrates a spectral efficiency comparison based on ACLR1between modified multiple layer overlay and square root raised cosine;

FIG. 35 illustrates a spectral efficiency comparison between modifiedmultiple layer overlay and square root raised cosine based on OBP;

FIG. 36 illustrates a spectral efficiency comparison based on ACLR1between modified multiple layer overlay and square root raised cosine;

FIG. 37 illustrates a spectral efficiency comparison based on OBPbetween modified multiple layer overlay and square root raised cosine;

FIG. 38 illustrates a block diagram of a baseband transmitter for a lowpass equivalent modified multiple layer overlay system;

FIG. 39 illustrates a block diagram of a baseband receiver for a lowpass equivalent modified multiple layer overlay system;

FIG. 40 illustrates the configuration of an optical fiber communicationsystem;

FIG. 41A illustrates a single mode fiber;

FIG. 41B illustrates multi-core fibers;

FIG. 41C illustrates multi-mode fibers;

FIG. 41D illustrates a hollow core fiber;

FIG. 42 illustrates the first six modes within a step index fiber;

FIG. 43 illustrates the classes of random perturbations within a fiber;

FIG. 44 illustrates the intensity patterns of first order groups withina vortex fiber;

FIGS. 45A and 45B illustrate index separation in first order modes of amulti-mode fiber;

FIG. 46 illustrates a free-space communication system;

FIG. 47 illustrates a block diagram of a free-space optics system usingorbital angular momentum and multi-level overlay modulation;

FIGS. 48A-48C illustrate the manner for multiplexing multiple datachannels into optical links to achieve higher data capacity;

FIG. 48D illustrates groups of concentric rings for a wavelength havingmultiple OAM valves;

FIG. 49 illustrates a WDM channel containing many orthogonal OAM beams;

FIG. 50 illustrates a node of a free-space optical system;

FIG. 51 illustrates a network of nodes within a free-space opticalsystem;

FIG. 52 illustrates a system for multiplexing between a free spacesignal and an RF signal;

FIG. 53 illustrates a block diagram of an OAM processing systemutilizing quantum key distribution;

FIG. 54 illustrates a basic quantum key distribution system;

FIG. 55 illustrates the manner in which two separate states are combinedinto a single conjugate pair within quantum key distribution;

FIG. 56 illustrates one manner in which 0 and 1 bits may be transmittedusing different basis within a quantum key distribution system;

FIG. 57 is a flow diagram illustrating the process for a transmittertransmitting a quantum key;

FIG. 58 illustrates the manner in which the receiver may receive anddetermine a shared quantum key;

FIG. 59 more particularly illustrates the manner in which a transmitterand receiver may determine a shared quantum key;

FIG. 60 is a flow diagram illustrating the process for determiningwhether to keep or abort a determined key;

FIG. 61 illustrates a functional block diagram of transmitter andreceiver utilizing a free-space quantum key distribution system;

FIG. 62 illustrates a network cloud-based quantum key distributionsystem;

FIG. 63 illustrates a high-speed single photon detector in communicationwith a plurality of users; and

FIG. 64 illustrates a nodal quantum key distribution network.

DETAILED DESCRIPTION

Referring now to the drawings, wherein like reference numbers are usedherein to designate like elements throughout, the various views andembodiments of system and method for communication using orbital angularmomentum with modulation are illustrated and described, and otherpossible embodiments are described. The figures are not necessarilydrawn to scale, and in some instances the drawings have been exaggeratedand/or simplified in places for illustrative purposes only. One ofordinary skill in the art will appreciate the many possible applicationsand variations based on the following examples of possible embodiments.

Referring now to the drawings, and more particularly to FIG. 1, whereinthere is illustrated two manners for increasing spectral efficiency of acommunications system. In general, there are basically two ways toincrease spectral efficiency 102 of a communications system. Theincrease may be brought about by signal processing techniques 104 in themodulation scheme or using multiple access technique. Additionally, thespectral efficiency can be increase by creating new Eigen channels 106within the electromagnetic propagation. These two techniques arecompletely independent of one another and innovations from one class canbe added to innovations from the second class. Therefore, thecombination of this technique introduced a further innovation.

Spectral efficiency 102 is the key driver of the business model of acommunications system. The spectral efficiency is defined in units ofbit/sec/hz and the higher the spectral efficiency, the better thebusiness model. This is because spectral efficiency can translate to agreater number of users, higher throughput, higher quality or some ofeach within a communications system.

Regarding techniques using signal processing techniques or multipleaccess techniques. These techniques include innovations such as TDMA,FDMA, CDMA, EVDO, GSM, WCDMA, HSPA and the most recent OFDM techniquesused in 4G WIMAX and LTE. Almost all of these techniques use decades-oldmodulation techniques based on sinusoidal Eigen functions called QAMmodulation. Within the second class of techniques involving the creationof new Eigen channels 106, the innovations include diversity techniquesincluding space and polarization diversity as well as multipleinput/multiple output (MIMO) where uncorrelated radio paths createindependent Eigen channels and propagation of electromagnetic waves.

Referring now to FIG. 2, the present communication system configurationintroduces two techniques, one from the signal processing techniques 104category and one from the creation of new eigen channels 106 categorythat are entirely independent from each other. Their combinationprovides a unique manner to disrupt the access part of an end to endcommunications system from twisted pair and cable to fiber optics, tofree space optics, to RF used in cellular, backhaul and satellite. Thefirst technique involves the use of a new signal processing techniqueusing new orthogonal signals to upgrade QAM modulation using nonsinusoidal functions. This is referred to as quantum level overlay (QLO)202. The second technique involves the application of newelectromagnetic wavefronts using a property of electromagnetic waves orphoton, called orbital angular momentum (QAM) 104. Application of eachof the quantum level overlay techniques 202 and orbital angular momentumapplication 204 uniquely offers orders of magnitude higher spectralefficiency 206 within communication systems in their combination.

With respect to the quantum level overlay technique 202, new eigenfunctions are introduced that when overlapped (on top of one anotherwithin a symbol) significantly increases the spectral efficiency of thesystem. The quantum level overlay technique 302 borrows from quantummechanics, special orthogonal signals that reduce the time bandwidthproduct and thereby increase the spectral efficiency of the channel.Each orthogonal signal is overlaid within the symbol acts as anindependent channel. These independent channels differentiate thetechnique from existing modulation techniques.

With respect to the application of orbital angular momentum 204, thistechnique introduces twisted electromagnetic waves, or light beams,having helical wave fronts that carry orbital angular momentum (OAM).Different OAM carrying waves/beams can be mutually orthogonal to eachother within the spatial domain, allowing the waves/beams to beefficiently multiplexed and demultiplexed within a communications link.OAM beams are interesting in communications due to their potentialability in special multiplexing multiple independent data carryingchannels.

With respect to the combination of quantum level overlay techniques 202and orbital angular momentum application 204, the combination is uniqueas the OAM multiplexing technique is compatible with otherelectromagnetic techniques such as wave length and polarization divisionmultiplexing. This suggests the possibility of further increasing systemperformance. The application of these techniques together in highcapacity data transmission disrupts the access part of an end to endcommunications system from twisted pair and cable to fiber optics, tofree space optics, to RF used in cellular/backhaul and satellites.

Each of these techniques can be applied independent of one another, butthe combination provides a unique opportunity to not only increasespectral efficiency, but to increase spectral efficiency withoutsacrificing distance or signal to noise ratios.

Using the Shannon Capacity Equation, a determination may be made ifspectral efficiency is increased. This can be mathematically translatedto more bandwidth. Since bandwidth has a value, one can easily convertspectral efficiency gains to financial gains for the business impact ofusing higher spectral efficiency. Also, when sophisticated forward errorcorrection (FEC) techniques are used, the net impact is higher qualitybut with the sacrifice of some bandwidth. However, if one can achievehigher spectral efficiency (or more virtual bandwidth), one cansacrifice some of the gained bandwidth for FEC and therefore higherspectral efficiency can also translate to higher quality.

Telecom operators and vendors are interested in increasing spectralefficiency. However, the issue with respect to this increase is thecost. Each technique at different layers of the protocol has a differentprice tag associated therewith. Techniques that are implemented at aphysical layer have the most impact as other techniques can besuperimposed on top of the lower layer techniques and thus increase thespectral efficiency further. The price tag for some of the techniquescan be drastic when one considers other associated costs. For example,the multiple input multiple output (MIMO) technique uses additionalantennas to create additional paths where each RF path can be treated asan independent channel and thus increase the aggregate spectralefficiency. In the MIMO scenario, the operator has other associated softcosts dealing with structural issues such as antenna installations, etc.These techniques not only have tremendous cost, but they have hugetiming issues as the structural activities take time and the achievingof higher spectral efficiency comes with significant delays which canalso be translated to financial losses.

The quantum level overlay technique 202 has an advantage that theindependent channels are created within the symbols without needing newantennas. This will have a tremendous cost and time benefit compared toother techniques. Also, the quantum layer overlay technique 202 is aphysical layer technique, which means there are other techniques athigher layers of the protocol that can all ride on top of the QLOtechniques 202 and thus increase the spectral efficiency even further.QLO technique 202 uses standard QAM modulation used in OFDM basedmultiple access technologies such as WIMAX or LTE. QLO technique 202basically enhances the QAM modulation at the transceiver by injectingnew signals to the I & Q components of the baseband and overlaying thembefore QAM modulation as will be more fully described herein below. Atthe receiver, the reverse procedure is used to separate the overlaidsignal and the net effect is a pulse shaping that allows betterlocalization of the spectrum compared to standard QAM or even the rootraised cosine. The impact of this technique is a significantly higherspectral efficiency.

Referring now more particularly to FIG. 3, there is illustrated ageneral overview of the manner for providing improved communicationbandwidth within various communication protocol interfaces 302, using acombination of multiple level overlay modulation 304 and the applicationof orbital angular momentum 306 to increase the number of communicationschannels.

The various communication protocol interfaces 302 may comprise a varietyof communication links, such as RF communication, wireline communicationsuch as cable or twisted pair connections, or optical communicationsmaking use of light wavelengths such as fiber-optic communications orfree-space optics. Various types of RF communications may include acombination of RF microwave or RF satellite communication, as well asmultiplexing between RF and free-space optics in real time.

By combining a multiple layer overlay modulation technique 304 withorbital angular momentum (OAM) technique 306, a higher throughput overvarious types of communication links 302 may be achieved. The use ofmultiple level overlay modulation alone without OAM increases thespectral efficiency of communication links 302, whether wired, optical,or wireless. However, with OAM, the increase in spectral efficiency iseven more significant.

Multiple overlay modulation techniques 304 provide a new degree offreedom beyond the conventional 2 degrees of freedom, with time T andfrequency F being independent variables in a two-dimensional notationalspace defining orthogonal axes in an information diagram. This comprisesa more general approach rather than modeling signals as fixed in eitherthe frequency or time domain. Previous modeling methods using fixed timeor fixed frequency are considered to be more limiting cases of thegeneral approach of using multiple level overlay modulation 304. Withinthe multiple level overlay modulation technique 304, signals may bedifferentiated in two-dimensional space rather than along a single axis.Thus, the information-carrying capacity of a communications channel maybe determined by a number of signals which occupy different time andfrequency coordinates and may be differentiated in a notationaltwo-dimensional space.

Within the notational two-dimensional space, minimization of the timebandwidth product, i.e., the area occupied by a signal in that space,enables denser packing, and thus, the use of more signals, with higherresulting information-carrying capacity, within an allocated channel.Given the frequency channel delta (Δf), a given signal transmittedthrough it in minimum time Δt will have an envelope described by certaintime-bandwidth minimizing signals. The time-bandwidth products for thesesignals take the form;

ΔtΔf=½(2n+1)

where n is an integer ranging from 0 to infinity, denoting the order ofthe signal.

These signals form an orthogonal set of infinite elements, where eachhas a finite amount of energy. They are finite in both the time domainand the frequency domain, and can be detected from a mix of othersignals and noise through correlation, for example, by match filtering.Unlike other wavelets, these orthogonal signals have similar time andfrequency forms.

The orbital angular momentum process 306 provides a twist to wave frontsof the electromagnetic fields carrying the data stream that may enablethe transmission of multiple data streams on the same frequency,wavelength, or other signal-supporting mechanism. This will increase thebandwidth over a communications link by allowing a single frequency orwavelength to support multiple eigen channels, each of the individualchannels having a different orthogonal and independent orbital angularmomentum associated therewith.

Referring now to FIG. 4, there is illustrated a further communicationimplementation technique using the above described techniques as twistedpairs or cables carry electrons (not photons). Rather than using each ofthe multiple level overlay modulation 304 and orbital angular momentumtechniques 306, only the multiple level overlay modulation 304 can beused in conjunction with a single wireline interface and, moreparticularly, a twisted pair communication link or a cable communicationlink 402. The operation of the multiple level overlay modulation 404, issimilar to that discussed previously with respect to FIG. 3, but is usedby itself without the use of orbital angular momentum techniques 306,and is used with either a twisted pair communication link or cableinterface communication link 402.

Referring now to FIG. 5, there is illustrated a general block diagramfor processing a plurality of data streams 502 for transmission in anoptical communication system. The multiple data streams 502 are providedto the multi-layer overlay modulation circuitry 504 wherein the signalsare modulated using the multi-layer overlay modulation technique. Themodulated signals are provided to orbital angular momentum processingcircuitry 506 which applies a twist to each of the wave fronts beingtransmitted on the wavelengths of the optical communication channel. Thetwisted waves are transmitted through the optical interface 508 over anoptical communications link such as an optical fiber or free spaceoptics communication system. FIG. 5 may also illustrate an RF mechanismwherein the interface 508 would comprise and RF interface rather than anoptical interface.

Referring now more particularly to FIG. 6, there is illustrated afunctional block diagram of a system for generating the orbital angularmomentum “twist” within a communication system, such as that illustratedwith respect to FIG. 3, to provide a data stream that may be combinedwith multiple other data streams for transmission upon a same wavelengthor frequency. Multiple data streams 602 are provided to the transmissionprocessing circuitry 600. Each of the data streams 602 comprises, forexample, an end to end link connection carrying a voice call or a packetconnection transmitting non-circuit switch packed data over a dataconnection. The multiple data streams 602 are processed bymodulator/demodulator circuitry 604. The modulator/demodulator circuitry604 modulates the received data stream 602 onto a wavelength orfrequency channel using a multiple level overlay modulation technique,as will be more fully described herein below. The communications linkmay comprise an optical fiber link, free-space optics link, RF microwavelink, RF satellite link, wired link (without the twist), etc.

The modulated data stream is provided to the orbital angular momentum(OAM) signal processing block 606. Each of the modulated data streamsfrom the modulator/demodulator 604 are provided a different orbitalangular momentum by the orbital angular momentum electromagnetic block606 such that each of the modulated data streams have a unique anddifferent orbital angular momentum associated therewith. Each of themodulated signals having an associated orbital angular momentum areprovided to an optical transmitter 608 that transmits each of themodulated data streams having a unique orbital angular momentum on asame wavelength. Each wavelength has a selected number of bandwidthslots B and may have its data transmission capability increase by afactor of the number of degrees of orbital angular momentum l that areprovided from the OAM electromagnetic block 606. The optical transmitter608 transmitting signals at a single wavelength could transmit B groupsof information. The optical transmitter 608 and OAM electromagneticblock 606 may transmit×B groups of information according to theconfiguration described herein.

In a receiving mode, the optical transmitter 608 will have a wavelengthincluding multiple signals transmitted therein having different orbitalangular momentum signals embedded therein. The optical transmitter 608forwards these signals to the OAM signal processing block 606, whichseparates each of the signals having different orbital angular momentumand provides the separated signals to the demodulator circuitry 604. Thedemodulation process extracts the data streams 602 from the modulatedsignals and provides it at the receiving end using the multiple layeroverlay demodulation technique.

Referring now to FIG. 7, there is provided a more detailed functionaldescription of the OAM signal processing block 606. Each of the inputdata streams are provided to OAM circuitry 702. Each of the OAMcircuitry 702 provides a different orbital angular momentum to thereceived data stream. The different orbital angular momentums areachieved by applying different currents for the generation of thesignals that are being transmitted to create a particular orbitalangular momentum associated therewith. The orbital angular momentumprovided by each of the OAM circuitries 702 are unique to the datastream that is provided thereto. An infinite number of orbital angularmomentums may be applied to different input data streams using manydifferent currents. Each of the separately generated data streams areprovided to a signal combiner 704, which combines the signals onto awavelength for transmission from the transmitter 706.

Referring now to FIG. 8, there is illustrated the manner in which theOAM processing circuitry 606 may separate a received signal intomultiple data streams. The receiver 802 receives the combined GAMsignals on a single wavelength and provides this information to a signalseparator 804. The signal separator 804 separates each of the signalshaving different orbital angular momentums from the received wavelengthand provides the separated signals to OAM de-twisting circuitry 806. TheOAM de-twisting circuitry 806 removes the associated OAM twist from eachof the associated signals and provides the received modulated datastream for further processing. The signal separator 804 separates eachof the received signals that have had the orbital angular momentumremoved therefrom into individual received signals. The individuallyreceived signals are provided to the receiver 802 for demodulationusing, for example, multiple level overlay demodulation as will be morefully described herein below.

FIG. 9 illustrates in a manner in which a single wavelength orfrequency, having two quanti-spin polarizations may provide an infinitenumber of twists having various orbital angular momentums associatedtherewith. The I axis represents the various quantized orbital angularmomentum states which may be applied to a particular signal at aselected frequency or wavelength. The symbol omega (ω) represents thevarious frequencies to which the signals of differing orbital angularmomentum may be applied. The top grid 902 represents the potentiallyavailable signals for a left handed signal polarization, while thebottom grid 904 is for potentially available signals having right handedpolarization.

By applying different orbital angular momentum states to a signal at aparticular frequency or wavelength, a potentially infinite number ofstates may be provided at the frequency or wavelength. Thus, the stateat the frequency Δω or wavelength 906 in both the left handedpolarization plane 902 and the right handed polarization plane 904 canprovide an infinite number of signals at different orbital angularmomentum states Δl. Blocks 908 and 910 represent a particular signalhaving an orbital angular momentum Δl at a frequency Δω or wavelength inboth the right handed polarization plane 904 and left handedpolarization plane 910, respectively. By changing to a different orbitalangular momentum within the same frequency Δω or wavelength 906,different signals may also be transmitted. Each angular momentum statecorresponds to a different determined current level for transmissionfrom the optical transmitter. By estimating the equivalent current forgenerating a particular orbital angular momentum within the opticaldomain and applying this current for transmission of the signals, thetransmission of the signal may be achieved at a desired orbital angularmomentum state.

Thus, the illustration of FIG. 9, illustrates two possible angularmomentums, the spin angular momentum, and the orbital angular momentum.The spin version is manifested within the polarizations of macroscopicelectromagnetism, and has only left and right hand polarizations due toup and down spin directions. However, the orbital angular momentumindicates an infinite number of states that are quantized. The paths aremore than two and can theoretically be infinite through the quantizedorbital angular momentum levels.

Using the orbital angular momentum state of the transmitted energysignals, physical information can be embedded within the radiationtransmitted by the signals. The Maxwell-Heaviside equations can berepresented as:

${\nabla{\cdot E}} = \frac{\rho}{ɛ_{0}}$${\nabla{\times E}} = {- \frac{\partial B}{\partial t}}$ ∇⋅B = 0${\nabla{\times B}} = {{ɛ_{0}\mu_{0}\frac{\partial E}{\partial t}} + {\mu_{0}{j\left( {t,x} \right)}}}$

where ∇ is the del operator, E is the electric field intensity and B isthe magnetic flux density. Using these equations, one can derive 23symmetries/conserved quantities from Maxwell's original equations.However, there are only ten well-known conserved quantities and only afew of these are commercially used. Historically if Maxwell's equationswhere kept in their original quaternion forms, it would have been easierto see the symmetries/conserved quantities, but when they were modifiedto their present vectorial form by Heaviside, it became more difficultto see such inherent symmetries in Maxwell's equations.

Maxwell's linear theory is of U(1) symmetry with Abelian commutationrelations. They can be extended to higher symmetry group SU(2) form withnon-Abelian commutation relations that address global (non-local inspace) properties. The Wu-Yang and Harmuth interpretation of Maxwell'stheory implicates the existence of magnetic monopoles and magneticcharges. As far as the classical fields are concerned, these theoreticalconstructs are pseudo-particle, or instanton. The interpretation ofMaxwell's work actually departs in a significant ways from Maxwell'soriginal intention. In Maxwell's original formulation, Faraday'selectrotonic states (the Aμ field) was central making them compatiblewith Yang-Mills theory (prior to Heaviside). The mathematical dynamicentities called solitons can be either classical or quantum, linear ornon-linear and describe EM waves. However, solitons are of SU(2)symmetry forms. In order for conventional interpreted classicalMaxwell's theory of U(1) symmetry to describe such entities, the theorymust be extended to SU(2) forms.

Besides the half dozen physical phenomena (that cannot be explained withconventional Maxwell's theory), the recently formulated Harmuth Ansatzalso address the incompleteness of Maxwell's theory. Harmuth amendedMaxwell's equations can be used to calculate EM signal velocitiesprovided that a magnetic current density and magnetic charge are addedwhich is consistent to Yang-Mills filed equations. Therefore, with thecorrect geometry and topology, the Aμ potentials always have physicalmeaning

The conserved quantities and the electromagnetic field can berepresented according to the conservation of system energy and theconservation of system linear momentum. Time symmetry, i.e. theconservation of system energy can be represented using Poynting'stheorem according to the equations:

$\begin{matrix}{H = {{\sum\limits_{i}{m_{i}\gamma_{i}c^{2}}} + {\frac{ɛ_{0}}{2}{\int_{\;}^{\;}{^{3}{x\left( {{E}^{2} + {c^{2}{B}^{2}}} \right)}}}}}} & {{Hamiltonian}\mspace{14mu} \left( {{total}\mspace{14mu} {energy}} \right)} \\{{\frac{U^{mech}}{t} + \frac{U^{em}}{t} + {\oint_{s^{\prime}}{{^{2}x^{\prime}}{\hat{n^{\prime}} \cdot S}}}} = 0} & {{conservation}\mspace{14mu} {of}\mspace{14mu} {energy}}\end{matrix}$

The space symmetry, i.e., the conservation of system linear momentumrepresenting the electromagnetic Doppler shift can be represented by theequations:

$\begin{matrix}{p = {{\sum\limits_{i}{m_{i}\gamma_{i}v_{i}}} + {ɛ_{0}{\int_{\;}^{\;}{^{3}{x\left( {E \times B} \right)}}}}}} & {{linear}\mspace{14mu} {momentum}} \\{{\frac{p^{mech}}{t} + \frac{p^{em}}{t} + {\oint_{s^{\prime}}{{^{2}x^{\prime}}{n^{\hat{\prime}} \cdot T}}}} = 0} & {{conservation}\mspace{14mu} {of}\mspace{14mu} {linear}\mspace{14mu} {momentum}}\end{matrix}$

The conservation of system center of energy is represented by theequation:

$R = {{\frac{1}{H}{\sum\limits_{i}{\left( {x_{i} - x_{0}} \right)m_{i}\gamma_{i}c^{2}}}} + {\frac{ɛ_{0}}{2H}{\int_{\;}^{\;}\; {{^{3}{x\left( {x - x_{0}} \right)}}\left( {{E^{2}} + {c^{2}{B^{2}}}} \right)}}}}$

Similarly, the conservation of system angular momentum, which gives riseto the azimuthal Doppler shift is represented by the equation:

${\frac{J^{mech}}{t} + \frac{J^{em}}{t} + {\oint\limits_{s^{\prime}}{{^{2}x^{\prime}}{{\hat{n}}^{\prime} \cdot M}}}} = 0$conservation  of  angular  momentum

For radiation beams in free space, the EM field angular momentum J^(em)can be separated into two parts:

J ^(em)=ε₀∫_(V′) d ³ x′(E×A)+ε₀∫_(V′) d ³ x′E _(i)[(x′−x ₀)×∇]A _(i)

For each singular Fourier mode in real valued representation:

$J^{em} = {{{- }\frac{ɛ_{0}}{2\omega}{\int_{V^{\prime}}^{\;}\ {^{3}{x^{\prime}\left( {E^{*} \times E} \right)}}}} - {\frac{ɛ_{0}}{2\omega}{\int_{V^{\prime}}^{\;}\ {{^{3}x^{\prime}}{E_{i}\left\lbrack {\left( {x^{\prime} - x_{0}} \right) \times \nabla} \right\rbrack}E_{i}}}}}$

The first part is the EM spin angular momentum S^(em), its classicalmanifestation is wave polarization. And the second part is the EMorbital angular momentum L^(em) its classical manifestation is wavehelicity. In general, both EM linear momentum P^(em), and EM angularmomentum J^(em)=L^(em)+S^(em) are radiated all the way to the far field.

By using Poynting theorem, the optical vorticity of the signals may bedetermined according to the optical velocity equation:

${{\frac{\partial U}{\partial t} + {\nabla{\cdot S}}} = 0},{{continuity}\mspace{14mu} {equation}}$

where S is the Poynting vector

${S = {\frac{1}{4}\left( {{E \times H^{*}} + {E^{*} \times H}} \right)}},$

and U is the energy density

${U = {\frac{1}{4}\left( {{ɛ{E}^{2}} + {\mu_{0}{H}^{2}}} \right)}},$

with E and H comprising the electric field and the magnetic field,respectively, and ε and μ₀ being the permittivity and the permeabilityof the medium, respectively. The optical vorticity V may then bedetermined by the curl of the optical velocity according to theequation:

$V = {{\nabla{\times v_{opt}}} = {\nabla{\times \left( \frac{{E \times H^{*}} + {E^{*} \times H}}{{ɛ{E}^{2}} + {\mu_{0}{H}^{2}}} \right)}}}$

Referring now to FIGS. 10A and 10B, there is illustrated the manner inwhich a signal and its associated Poynting vector in a plane wavesituation. In the plane wave situation illustrated generally at 1002,the transmitted signal may take one of three configurations. When theelectric field vectors are in the same direction, a linear signal isprovided, as illustrated generally at 1004. Within a circularpolarization 1006, the electric field vectors rotate with the samemagnitude. Within the elliptical polarization 1008, the electric fieldvectors rotate but have differing magnitudes. The Poynting vectorremains in a constant direction for the signal configuration to FIG. 10Aand always perpendicular to the electric and magnetic fields. Referringnow to FIG. 10B, when a unique orbital angular momentum is applied to asignal as described here and above, the Poynting vector S 1010 willspiral about the direction of propagation of the signal. This spiral maybe varied in order to enable signals to be transmitted on the samefrequency as described herein.

FIGS. 11A through 11C illustrate the differences in signals havingdifferent helicity (i.e., orbital angular momentums). Each of thespiraling Poynting vectors associated with the signals 1102, 1104, and1106 provide a different shaped signal. Signal 1102 has an orbitalangular momentum of +1, signal 1104 has an orbital angular momentum of+3, and signal 1106 has an orbital angular momentum of −4. Each signalhas a distinct angular momentum and associated Poynting vector enablingthe signal to be distinguished from other signals within a samefrequency. This allows differing type of information to be transmittedon the same frequency, since these signals are separately detectable anddo not interfere with each other (Eigen channels).

FIG. 11D illustrates the propagation of Poynting vectors for variousEigen modes. Each of the rings 1120 represents a different Eigen mode ortwist representing a different orbital angular momentum within the samefrequency. Each of these rings 1120 represents a different orthogonalchannel. Each of the Eigen modes has a Poynting vector 1122 associatedtherewith.

Topological charge may be multiplexed to the frequency for either linearor circular polarization. In case of linear polarizations, topologicalcharge would be multiplexed on vertical and horizontal polarization. Incase of circular polarization, topological charge would multiplex onleft hand and right hand circular polarizations. The topological chargeis another name for the helicity index “I” or the amount of twist or OAMapplied to the signal. The helicity index may be positive or negative.In RF, different topological charges can be created and muxed togetherand de-muxed to separate the topological charges.

The topological charges t s can be created using Spiral Phase Plates(SPPs) as shown in FIG. 11E using a proper material with specific indexof refraction and ability to machine shop or phase mask, hologramscreated of new materials or a new technique to create an RF version ofSpatial Light Modulator (SLM) that does the twist of the RF waves (asopposed to optical beams) by adjusting voltages on the device resultingin twisting of the RF waves with a specific topological charge. SpiralPhase plates can transform a RF plane wave (l=0) to a twisted RF wave ofa specific helicity (i.e. l=+1).

Cross talk and multipath interference can be corrected using RFMultiple-Input-Multiple-Output (MIMO). Most of the channel impairmentscan be detected using a control or pilot channel and be corrected usingalgorithmic techniques (closed loop control system).

As described previously with respect to FIG. 5, each of the multipledata streams applied within the processing circuitry has a multiplelayer overlay modulation scheme applied thereto.

Referring now to FIG. 12, the reference number 1200 generally indicatesan embodiment of a multiple level overlay (MLO) modulation system,although it should be understood that the term MLO and the illustratedsystem 1200 are examples of embodiments. The MLO system may comprise onesuch as that disclosed in U.S. Pat. No. 8,503,546 entitled MultipleLayer Overlay Modulation which is incorporated herein by reference. Inone example, the modulation system 1200 would be implemented within themultiple level overlay modulation box 504 of FIG. 5. System 1200 takesas input an input data stream 1201 from a digital source 1202, which isseparated into three parallel, separate data streams, 1203A-1203C, oflogical 1s and 0s by input stage demultiplexer (DEMUX) 1004. Data stream1001 may represent a data file to be transferred, or an audio or videodata stream. It should be understood that a greater or lesser number ofseparated data streams may be used. In some of the embodiments, each ofthe separated data streams 1203A-1203C has a data rate of 1/N of theoriginal rate, where N is the number of parallel data streams. In theembodiment illustrated in FIG. 12, N is 3.

Each of the separated data streams 1203A-1203C is mapped to a quadratureamplitude modulation (QAM) symbol in an M-QAM constellation, forexample, 16 QAM or 64 QAM, by one of the QAM symbol mappers 1205A-C. TheQAM symbol mappers 1205A-C are coupled to respective outputs of DEMUX1204, and produced parallel in phase (I) 1206A, 1208A, and 1210A andquadrature phase (Q) 1206B, 12089, and 1210B data streams at discretelevels. For example, in 64 QAM, each I and Q channel uses 8 discretelevels to transmit 3 bits per symbol. Each of the three I and Q pairs,1206A-1206B, 1208A-1208B, and 1210A-1210B, is used to weight the outputof the corresponding pair of function generators 1207A-1207B,1209A-1209B, and 1211A-1211B, which in some embodiments generate signalssuch as the modified Hermite polynomials described above and weightsthem based on the amplitude value of the input symbols. This provides 2Nweighted or modulated signals, each carrying a portion of the dataoriginally from income data stream 1201, and is in place of modulatingeach symbol in the I and Q pairs, 1206A-1206B, 1208A-1208B, and1210A-1210B with a raised cosine filter, as would be done for a priorart QAM system. In the illustrated embodiment, three signals are used,SH0, SH1, and SH2, which correspond to modifications of H0, H1, and H2,respectively, although it should be understood that different signalsmay be used in other embodiments.

The weighted signals are not subcarriers, but rather are sublayers of amodulated carrier, and are combined, superimposed in both frequency andtime, using summers 1212 and 1216, without mutual interference in eachof the I and Q dimensions, due to the signal orthogonality. Summers 1212and 1216 act as signal combiners to produce composite signals 1213 and1217. The weighted orthogonal signals are used for both I and Qchannels, which have been processed equivalently by system 1200, and aresummed before the QAM signal is transmitted. Therefore, although neworthogonal functions are used, some embodiments additionally use QAM fortransmission. Because of the tapering of the signals in the time domain,as will be shown in FIGS. 16A through 16K, the time domain waveform ofthe weighted signals will be confined to the duration of the symbols.Further, because of the tapering of the special signals and frequencydomain, the signal will also be confined to frequency domain, minimizinginterface with signals and adjacent channels.

The composite signals 1213 and 1217 are converted to analogue signals1215 and 1219 using digital to analogue converters 1214 and 1218, andare then used to modulate a carrier signal at the frequency of localoscillator (LO) 1220, using modulator 1221. Modulator 1221 comprisesmixers 1222 and 1224 coupled to DACs 1214 and 1218, respectively. Ninetydegree phase shifter 1223 converts the signals from LO 1220 into a Qcomponent of the carrier signal. The output of mixers 1222 and 1224 aresummed in summer 1225 to produce output signals 1226.

MLO can be used with a variety of transport mediums, such as wire,optical, and wireless, and may be used in conjunction with QAM. This isbecause MLO uses spectral overlay of various signals, rather thanspectral overlap. Bandwidth utilization efficiency may be increased byan order of magnitude, through extensions of available spectralresources into multiple layers. The number of orthogonal signals isincreased from 2, cosine and sine, in the prior art, to a number limitedby the accuracy and jitter limits of generators used to produce theorthogonal polynomials. In this manner, MLO extends each of the I and Qdimensions of QAM to any multiple access techniques such as GSM, codedivision multiple access (CDMA), wide band CDMA (WCDMA), high speeddownlink packet access (HSPDA), evolution-data optimized (EV-DO),orthogonal frequency division multiplexing (OFDM), world-wideinteroperability for microwave access (WIMAX), and long term evolution(LTE) systems. MLO may be further used in conjunction with othermultiple access (MA) schemes such as frequency division duplexing (FDD),time division duplexing (TDD), frequency division multiple access(FDMA), and time division multiple access (TDMA). Overlaying individualorthogonal signals over the same frequency band allows creation of avirtual bandwidth wider than the physical bandwidth, thus adding a newdimension to signal processing. This modulation is applicable to twistedpair, cable, fiber optic, satellite, broadcast, free-space optics, andall types of wireless access. The method and system are compatible withmany current and future multiple access systems, including EV-DO, UMB,WIMAX, WCDMA (with or without), multimedia broadcast multicast service(MBMS)/multiple input multiple output (MIMO), HSPA evolution, and LTE.

Referring now to FIG. 13, an MLO demodulator 1300 is illustrated,although it should be understood that the term MLO and the illustratedsystem 1300 are examples of embodiments. The modulator 1300 takes asinput an MLO signal 1126 which may be similar to output signal 1226 fromsystem 1200. Synchronizer 1327 extracts phase information, which isinput to local oscillator 1320 to maintain coherence so that themodulator 1321 can produce base band to analogue I signal 1315 and Qsignal 1319. The modulator 1321 comprises mixers 1322 and 1324, which,coupled to OL 1320 through 90 degree phase shifter 1323. I signal 1315is input to each of signal filters 1307A, 1309A, and 1311A, and Q signal1319 is input to each of signal filters 1307B, 1309B, and 1311B. Sincethe orthogonal functions are known, they can be separated usingcorrelation or other techniques to recover the modulated data.Information in each of the I and Q signals 1315 and 1319 can beextracted from the overlapped functions which have been summed withineach of the symbols because the functions are orthogonal in acorrelative sense.

In some embodiments, signal filters 1307A-1307B, 1309A-1309B, and1311A-1311B use locally generated replicas of the polynomials as knownsignals in match filters. The outputs of the match filters are therecovered data bits, for example, equivalence of the QAM symbols1306A-1306B, 1308A-1308B, and 1310A-1310B of system 1300. Signal filters1307A-1307B, 1309A-1309B, and 1311A-1311B produce 2n streams of n, I,and Q signal pairs, which are input into demodulators 1328-1333.Demodulators 1328-1333 integrate the energy in their respective inputsignals to determine the value of the QAM symbol, and hence the logical1s and 0s data bit stream segment represented by the determined symbol.The outputs of the modulators 1328-1333 are then input into multiplexers(MUXs) 1305A-1305C to generate data streams 1303A-1303C. If system 1300is demodulating a signal from system 1200, data streams 1303A-1303Ccorrespond to data streams 1203A-1203C. Data streams 1303A-1303C aremultiplexed by MUX 1304 to generate data output stream 1301. In summary,MLO signals are overlayed (stacked) on top of one another on transmitterand separated on receiver.

MLO may be differentiated from CDMA or OFDM by the manner in whichorthogonality among signals is achieved. MLO signals are mutuallyorthogonal in both time and frequency domains, and can be overlaid inthe same symbol time bandwidth product. Orthogonality is attained by thecorrelation properties, for example, by least sum of squares, of theoverlaid signals. In comparison, CDMA uses orthogonal interleaving ordisplacement of signals in the time domain, whereas OFDM uses orthogonaldisplacement of signals in the frequency domain.

Bandwidth efficiency may be increased for a channel by assigning thesame channel to multiple users. This is feasible if individual userinformation is mapped to special orthogonal functions. CDMA systemsoverlap multiple user information and views time intersymbol orthogonalcode sequences to distinguish individual users, and OFDM assigns uniquesignals to each user, but which are not overlaid, are only orthogonal inthe frequency domain. Neither CDMA nor OFDM increases bandwidthefficiency. CDMA uses more bandwidth than is necessary to transmit datawhen the signal has a low signal to noise ratio (SNR). OFDM spreads dataover many subcarriers to achieve superior performance in multipathradiofrequency environments. OFDM uses a cyclic prefix OFDM to mitigatemultipath effects and a guard time to minimize intersymbol interference(ISI), and each channel is mechanistically made to behave as if thetransmitted waveform is orthogonal. (Sync function for each subcarrierin frequency domain.)

In contrast, MLO uses a set of functions which effectively form analphabet that provides more usable channels in the same bandwidth,thereby enabling high bandwidth efficiency. Some embodiments of MLO donot require the use of cyclic prefixes or guard times, and therefore,outperforms OFDM in spectral efficiency, peak to average power ratio,power consumption, and requires fewer operations per bit. In addition,embodiments of MLO are more tolerant of amplifier nonlinearities thanare CDMA and OFDM systems.

FIG. 14 illustrates an embodiment of an MLO transmitter system 1400,which receives input data stream 1401. System 1400 represents amodulator/controller 1401, which incorporates equivalent functionalityof DEMUX 1204, QAM symbol mappers 1205A-C, function generators1207A-1207B, 1209A-1209B, and 1211A-1211B, and summers 1212 and 1216 ofsystem 1200, shown in FIG. 12. However, it should be understood thatmodulator/controller 1401 may use a greater or lesser quantity ofsignals than the three illustrated in system 1200. Modulator/controller1401 may comprise an application specific integrated circuit (ASIC), afield programmable gate array (FPGA), and/or other components, whetherdiscrete circuit elements or integrated into a single integrated circuit(IC) chip.

Modulator/controller 1401 is coupled to DACs 1404 and 1407,communicating a 10 bit I signal 1402 and a 10 bit Q signal 1405,respectively. In some embodiments, I signal 1402 and Q signal 1405correspond to composite signals 1213 and 1217 of system 1200. It shouldbe understood, however, that the 10 bit capacity of I signal 1402 and Qsignal 1405 is merely representative of an embodiment. As illustrated,modulator/controller 1401 also controls DACs 1404 and 1407 using controlsignals 1403 and 1406, respectively. In some embodiments, DACs 1404 and1407 each comprise an AD5433, complementary metal oxide semiconductor(CMOS) 10 bit current output DAC, In some embodiments, multiple controlsignals are sent to each of DACs 1404 and 1407.

DACs 1404 and 1407 output analogue signals 1215 and 1219 to quadraturemodulator 1221, which is coupled to LO 1220. The output of modulator1220 is illustrated as coupled to a transmitter 1408 to transmit datawirelessly, although in some embodiments, modulator 1221 may be coupledto a fiber-optic modem, a twisted pair, a coaxial cable, or othersuitable transmission media.

FIG. 15 illustrates an embodiment of an MLO receiver system 1500 capableof receiving and demodulating signals from system 1400. System 1500receives an input signal from a receiver 1508 that may comprise inputmedium, such as RF, wired or optical. The modulator 1321 driven by LO1320 converts the input to baseband I signal 1315 and Q signal 1319. Isignal 1315 and Q signal 1319 are input to analogue to digital converter(ADC) 1509.

ADC 1509 outputs 10 bit signal 1510 to demodulator/controller 1501 andreceives a control signal 1512 from demodulator/controller 1501.Demodulator/controller 1501 may comprise an application specificintegrated circuit (ASIC), a field programmable gate array (FPGA),and/or other components, whether discrete circuit elements or integratedinto a single integrated circuit (IC) chip. Demodulator/controller 1501correlates received signals with locally generated replicas of thesignal set used, in order to perform demodulation and identify thesymbols sent. Demodulator/controller 1501 also estimates frequencyerrors and recovers the data clock, which is used to read data from theADC 1509. The clock timing is sent back to ADC 1509 using control signal1512, enabling ADC 1509 to segment the digital I and Q signals 1315 and1319. In some embodiments, multiple control signals are sent bydemodulator/controller 1501 to ADC 1509. Demodulator/controller 1501also outputs data signal 1301.

Hermite polynomials are a classical orthogonal polynomial sequence,which are the Eigenstates of a quantum harmonic oscillator. Signalsbased on Hermite polynomials possess the minimal time-bandwidth productproperty described above, and may be used for embodiments of MLOsystems. However, it should be understood that other signals may also beused, for example orthogonal polynomials such as Jacobi polynomials,Gegenbauer polynomials, Legendre polynomials, Chebyshev polynomials, andLaguerre polynomials. Q-functions are another class of functions thatcan be employed as a basis for MLO signals.

In quantum mechanics, a coherent state is a state of a quantum harmonicoscillator whose dynamics most closely resemble the oscillating behaviorof a classical harmonic oscillator system. A squeezed coherent state isany state of the quantum mechanical Hilbert space, such that theuncertainty principle is saturated. That is, the product of thecorresponding two operators takes on its minimum value. In embodimentsof an MLO system, operators correspond to time and frequency domainswherein the time-bandwidth product of the signals is minimized. Thesqueezing property of the signals allows scaling in time and frequencydomain simultaneously, without losing mutual orthogonality among thesignals in each layer. This property enables flexible implementations ofMLO systems in various communications systems.

Because signals with different orders are mutually orthogonal, they canbe overlaid to increase the spectral efficiency of a communicationchannel. For example, when n=0, the optimal baseband signal will have atime-bandwidth product of ½, which is the Nyquist Inter-SymbolInterference (ISI) criteria for avoiding ISI. However, signals withtime-bandwidth products of 3/2, 5/2, 7/2, and higher, can be overlaid toincrease spectral efficiency.

An embodiment of an MLO system uses functions based on modified Hermitepolynomials, 4n, and are defined by:

${\psi_{n}\left( {t,\xi} \right)} = {\frac{\left( {\tanh \; \xi} \right)^{n/2}}{2^{n/2}\left( {{n!}\cosh \; \xi} \right)^{1/2}}^{\frac{1}{2}{t^{2}{\lbrack{1 - {\tanh \; \xi}}\rbrack}}}{H_{n}\left( \frac{t}{\sqrt{2\; \cosh \; \xi \; \sinh \; \xi}} \right)}}$

where t is time, and ξ is a bandwidth utilization parameter. Plots ofΨ_(n) for n ranging from 0 to 9, along with their Fourier transforms(amplitude squared), are shown in FIGS. 5A-5K. The orthogonality ofdifferent orders of the functions may be verified by integrating:

∫∫ψ_(n)(t,ξ)ψ_(m)(t,ξ)dtdξ

The Hermite polynomial is defined by the contour integral:

${{H_{n}(z)} = {\frac{n!}{2{\pi }}{\oint{^{{- t^{2}} + {2t\; 2}}t^{{- n} - 1}{t}}}}},$

where the contour encloses the origin and is traversed in acounterclockwise direction. Hermite polynomials are described inMathematical Methods for Physicists, by George Arfken, for example onpage 416, the disclosure of which is incorporated by reference.

FIGS. 16A-16K illustrate representative MLO signals and their respectivespectral power densities based on the modified Hermite polynomials Ψ_(n)for n ranging from 0 to 9. FIG. 16A shows plots 1601 and 1604. Plot 1601comprises a curve 1627 representing Ψ₀ plotted against a time axis 1602and an amplitude axis 1603. As can be seen in plot 1601, curve 1627approximates a Gaussian curve. Plot 1604 comprises a curve 1637representing the power spectrum of Ψ₀ plotted against a frequency axis1605 and a power axis 1606. As can be seen in plot 1604, curve 1637 alsoapproximates a Gaussian curve. Frequency domain curve 1607 is generatedusing a Fourier transform of time domain curve 1627. The units of timeand frequency on axis 1602 and 1605 are normalized for basebandanalysis, although it should be understood that since the time andfrequency units are related by the Fourier transform, a desired time orfrequency span in one domain dictates the units of the correspondingcurve in the other domain. For example, various embodiments of MLOsystems may communicate using symbol rates in the megahertz (MHz) orgigahertz (GHz) ranges and the non-0 duration of a symbol represented bycurve 1627, i.e., the time period at which curve 1627 is above 0 wouldbe compressed to the appropriate length calculated using the inverse ofthe desired symbol rate. For an available bandwidth in the megahertzrange, the non-0 duration of a time domain signal will be in themicrosecond range.

FIGS. 16B-16J show plots 1607-1624, with time domain curves 1628-1636representing Ψ₁ through Ψ₉, respectively, and their correspondingfrequency domain curves 1638-1646. As can be seen in FIGS. 16A-16J, thenumber of peaks in the time domain plots, whether positive or negative,corresponds to the number of peaks in the corresponding frequency domainplot. For example, in plot 1623 of FIG. 16J, time domain curve 1636 hasfive positive and five negative peaks. In corresponding plot 1624therefore, frequency domain curve 1646 has ten peaks.

FIG. 16K shows overlay plots 1625 and 1626, which overlay curves1627-1636 and 1637-1646, respectively. As indicated in plot 1625, thevarious time domain curves have different durations. However, in someembodiments, the non-zero durations of the time domain curves are ofsimilar lengths. For an MLO system, the number of signals usedrepresents the number of overlays and the improvement in spectralefficiency. It should be understood that, while ten signals aredisclosed in FIGS. 16A-16K, a greater or lesser quantity of signals maybe used, and that further, a different set of signals, rather than theΨ_(n) signals plotted, may be used.

MLO signals used in a modulation layer have minimum time-bandwidthproducts, which enable improvements in spectral efficiency, and arequadratically integrable. This is accomplished by overlaying multipledemultiplexed parallel data streams, transmitting them simultaneouslywithin the same bandwidth. The key to successful separation of theoverlaid data streams at the receiver is that the signals used withineach symbols period are mutually orthogonal. MLO overlays orthogonalsignals within a single symbol period. This orthogonality prevents ISIand inter-carrier interference (ICI).

Because MLO works in the baseband layer of signal processing, and someembodiments use QAM architecture, conventional wireless techniques foroptimizing air interface, or wireless segments, to other layers of theprotocol stack will also work with MLO. Techniques such as channeldiversity, equalization, error correction coding, spread spectrum,interleaving and space-time encoding are applicable to MLO. For example,time diversity using a multipath-mitigating rake receiver can also beused with MLO. MLO provides an alternative for higher order QAM, whenchannel conditions are only suitable for low order QAM, such as infading channels. MLO can also be used with CDMA to extend the number oforthogonal channels by overcoming the Walsh code limitation of CDMA. MLOcan also be applied to each tone in an OFDM signal to increase thespectral efficiency of the OFDM systems.

Embodiments of MLO systems amplitude modulate a symbol envelope tocreate sub-envelopes, rather than sub-carriers. For data encoding, eachsub-envelope is independently modulated according to N-QAM, resulting ineach sub-envelope independently carrying information, unlike OFDM.Rather than spreading information over many sub-carriers, as is done inOFDM, for MLO, each sub-envelope of the carrier carries separateinformation. This information can be recovered due to the orthogonalityof the sub-envelopes defined with respect to the sum of squares overtheir duration and/or spectrum. Pulse train synchronization or temporalcode synchronization, as needed for CDMA, is not an issue, because MLOis transparent beyond the symbol level. MLO addresses modification ofthe symbol, but since CDMA and TDMA are spreading techniques of multiplesymbol sequences over time. MLO can be used along with CDMA and TDMA.

FIG. 17 illustrates a comparison of MLO signal widths in the time andfrequency domains. Time domain envelope representations 1701-1703 ofsignals SH0-SH3 are illustrated as all having a duration T_(S). SH0-SH3may represent PSI₀-PSI₂, or may be other signals. The correspondingfrequency domain envelope representations are 1705-1707, respectively.SH0 has a bandwidth BW, SH1 has a bandwidth three times BW, and SH2 hasa bandwidth of 5BW, which is five times as great as that of SH0. Thebandwidth used by an MLO system will be determined, at least in part, bythe widest bandwidth of any of the signals used. If each layer uses onlya single signal type within identical time windows, the spectrum willnot be fully utilized, because the lower order signals will use less ofthe available bandwidth than is used by the higher order signals.

FIG. 18 illustrates a spectral alignment of MLO signals that accountsfor the differing bandwidths of the signals, and makes spectral usagemore uniform, using SH0-SH3. Blocks 18014804 are frequency domain blocksof an OFDM signal with multiple subcarriers. Block 1803 is expanded toshow further detail. Block 1803 comprises a first layer 1803 x comprisedof multiple SH0 envelopes 1803 a-1803 o. A second layer 1803 y of SH1envelopes 1803 p-1803 t has one third the number of envelopes as thefirst layer. In the illustrated example, first layer 1803 x has 15 SH0envelopes, and second layer 1803 y has five SH1 envelopes. This isbecause, since the SH1 bandwidth envelope is three times as wide as thatof SH0, 15 SH0 envelopes occupy the same spectral width as five SH1envelopes. The third layer 1803 z of block 1803 comprises three SH2envelopes 1803 u-1803 w, because the SH2 envelope is five times thewidth of the SH0 envelope.

The total required bandwidth for such an implementation is a multiple ofthe least common multiple of the bandwidths of the MLO signals. In theillustrated example, the least common multiple of the bandwidth requiredfor SH0, SH1, and SH2 is 15BW, which is a block in the frequency domain.The OFDM-MLO signal can have multiple blocks, and the spectralefficiency of this illustrated implementation is proportional to(15+5+3)/15.

FIG. 19 illustrates another spectral alignment of MLO signals, which maybe used alternatively to alignment scheme shown in FIG. 18. In theembodiment illustrated in FIG. 19, the OFDM-MLO implementation stacksthe spectrum of SH0, SH1, and SH2 in such a way that the spectrum ineach layer is utilized uniformly. Layer 1900A comprises envelopes1901A-1901D, which includes both SH0 and SH2 envelopes. Similarly, layer1900C, comprising envelopes 1903A-1903D, includes both. SH0 and SH2envelopes. Layer 1900B, however, comprising envelopes 1902A-1902D,includes only SH1 envelopes. Using the ratio of envelope sizes describedabove, it can be easily seen that BW+5BW=3BW+3BW. Thus, for each SH0envelope in layer 1900A, there is one SH2 envelope also in layer 19000and two SH1 envelopes in layer 1900B.

Three Scenarios Compared:

1) MLO with 3 Layers defined by:

${{f_{0}(t)} = {W_{0}^{- \frac{t^{2}}{4}}}},{W_{0} = 0.6316}$${{f_{1}(t)} = {W_{1}t\; ^{- \frac{t^{2}}{4}}}},{W_{1} \approx 0.6316}$${{f_{2}(t)} = {{W_{2}\left( {t^{2} - 1} \right)}^{- \frac{t^{2}}{4}}}},{W_{2} \approx 0.4466}$

(The current FPGA implementation uses the truncation interval of [−6,6].)2) Conventional scheme using rectangular pulse3) Conventional scheme using a square-root raised cosine (SRRC) pulsewith a roll-off factor of 0.5

For MLO pulses and SRRC pulse, the truncation interval is denoted by[−t1, t1] in the following figures. For simplicity, we used the MLOpulses defined above, which can be easily scaled in time to get thedesired time interval (say micro-seconds or nano-seconds). For the SRRCpulse, we fix the truncation interval of [−3T, 3T] where T is the symbolduration for all results presented in this document.

Bandwidth Efficiency

The X-dB bounded power spectral density bandwidth is defined as thesmallest frequency interval outside which the power spectral density(PSD) is X dB below the maximum value of the PSD. The X-dB can beconsidered as the out-of-band attenuation.

The bandwidth efficiency is expressed in Symbols per second per Hertz.The bit per second per Hertz can be obtained by multiplying the symbolsper second per Hertz with the number of bits per symbol (i.e.,multiplying with log 2 M for M-ary QAM).

Truncation of MLO pulses introduces inter-layer interferences (ILI).However, the truncation interval of [−6, 6] yields negligible ILI while[−4, 4] causes slight tolerable ILI.

The bandwidth efficiency of MLO may be enhanced by allowing inter-symbolinterference (ISI). To realize this enhancement, designing transmitterside parameters as well as developing receiver side detection algorithmsand error performance evaluation can be performed.

Referring now to FIG. 20, there is illustrated the power spectraldensity of each layer SH0-SH2 within MLO and also for the combined threelayer MLO. 2002 illustrates the power spectral density of the SH0 layer;2004 illustrates the power spectral density of the SH1 layer; 2006illustrates the power spectral density of the SH2 layer, and 2008illustrates the combined power spectral density of each layer.

Referring now to FIG. 21, there is illustrated the power spectraldensity of each layer as well as the power spectral density of thecombined three layer in a log scale. 2102 represents the SH0 layer. 2104represents the SH1 layer. 2106 represents the SH2 layer. 2108 representsthe combined layers.

Referring now to FIG. 22, there is a bandwidth efficiency comparisonversus out of band attenuation (X-dB) where quantum level overlay pulsetruncation interval is [−6,6] and the symbol rate is 1/6. Referring alsoto FIG. 23, there is illustrated the bandwidth efficiency comparisonversus out of band attenuation (X-dB) where quantum level overlay pulsetruncation interval is [−6,6] and the symbol rate is 1/4.

The QLO signals are generated from the Physicist's special Hermitefunctions:

${{f_{n}\left( {t,\alpha} \right)} = {\sqrt{\frac{\alpha}{\sqrt{\pi}{n!}2^{n}}}{H_{n}\left( {\alpha \; t} \right)}^{- \frac{\alpha^{2}t^{2}}{2}}}},{\alpha > 0}$

Note that the initial hardware implementation is using

$\alpha = \frac{1}{\sqrt{2}}$

and for consistency with his part,

$\alpha = \frac{1}{\sqrt{2}}$

is used in all figures related to the spectral efficiency.

Let the low-pass-equivalent power spectral density (PSD) of the combinedQLO signals be X(f) and its bandwidth be B. Here the bandwidth isdefined by one of the following criteria,

ACLR1 (First Adjacent Channel Leakage Ratio) in dBc equals:

${{ACLR}\; 1} = \frac{\int_{B/2}^{3{B/2}}{{X(f)}{f}}}{\int_{- \infty}^{\infty}{{X(f)}{f}}}$

ACLR2 (Second Adjacent Channel Leakage Ratio) in dBc equals:

${{ACLR}\; 2} = \frac{\int_{3{B/2}}^{5{B/2}}{{X(f)}{f}}}{\int_{- \infty}^{\infty}{{X(f)}{f}}}$

Out-of-Band Power to Total Power Ratio is:

$\frac{2{\int_{B/2}^{\infty}{{X(f)}{f}}}}{\int_{- \infty}^{\infty}{{X(f)}{f}}}$

The Band-Edge PSD in dBc/100 kHz equals:

$\frac{\int_{B/2}^{\frac{B}{2} + 10^{5}}{{X(f)}{f}}}{\int_{- \infty}^{\infty}{{X(f)}{f}}}$

Referring now to FIG. 24 there is illustrated a performance comparisonusing ACLR1 and ACLR2 for both a square root raised cosine scheme and amultiple layer overlay scheme. Line 2402 illustrates the performance ofa square root raised cosine 2402 using ACLR1 versus an MLO 2404 usingACLR1. Additionally, a comparison between a square root raised cosine2406 using ACLR2 versus MLO 2408 using ACLR2 is illustrated. Table Aillustrates the performance comparison using ACLR.

TABLE A Criteria: ACLR3 ≦ −30 dBc pear bandwidth Spectral EfficiencyACLR2 ≦ −43 dBc pear bandwidth (SymboL/sec/Hz) Gain SRRC [−8T, 8T] β =0.22 0.8765 1.0 Symbol Duration N Layers (Tmol) QLO N = 3 Tmol = 4 1.1331.2926 [−8, 8] N = 4 Tmol = 5 1.094 1.2481 Tmol = 4 1.367 1.5596 N = 10Tmol = 8 1.185 1.3520 Tmol = 7 1.355 1.5459 Tmol = 6 1.580 1.8026 Tmol =5 1.896 2.1631 Tmol = 4 2.371 2.7051

Referring now to FIG. 25, there is illustrated a performance comparisonbetween a square root raised cosine 2502 and a MLO 2504 usingout-of-band power. Referring now also to Table B, there is illustrated amore detailed comparison of the performance using out-of-band power.

TABLE B Table 3: Performance Comparison Using Out-of-Band PowerCriterion: Out-of-band Power/ Spectral Efficiency Total Power ≦ −30 dB(SymboL/sec/Hz) Gain SRRC [−8T, 8T] β = 0.22 0.861 1.0 Symbol Duration NLayers (Tmol) QLO N = 3 Tmol = 4 1.080 1.2544 [−8, 8] N = 4 Tmol = 51.049 1.2184 Tmol = 4 1.311 1.5226 N = 10 Tmol = 8 1.152 1.3380 Tmol = 71.317 1.5296 Tmol = 6 1.536 1.7840 Tmol = 5 1.844 2.1417 Tmol = 4 2.3052.6771

Referring now to FIG. 26, there is further provided a performancecomparison between a square root raised cosine 2602 and a MLO 2604 usingband-edge PSD. A more detailed illustration of the performancecomparison is provided in Table C.

TABLE C Table 4: Performance Comparison Using Band-Edge PSD Criterion:Spectral Efficiency Hand-Edge PSD = −50 dBc/100 kHz (Symbol/sec/Hz) GainSRRC [−8T, 8T] β = 0.22 0.810 1.0 Symbol Duration N Layers (Tmol) QLO N= 3 Tmol = 4 0.925 1.1420 [−8, 8] N = 4 Tmol = 5 0.912 1.1259 Tmol = 41.14 1.4074 N = 10 Tmol = 8 1.049 1.2951 Tmol = 7 1.198 1.4790 Tmol = 61.398 1.7259 Tmol = 5 1.678 2.0716 Tmol = 4 2.097 2.5889

[01.65] Referring now to FIGS. 27 and 28, there are more particularlyillustrated the transmit subsystem (FIG. 27) and the receiver subsystem(FIG. 28). The transceiver is realized using basic building blocksavailable as Commercially Off The Shelf products. Modulation,demodulation and Special Hermite correlation and de-correlation areimplemented on a FPGA board. The FPGA board 2802 at the receiver 2800estimated the frequency error and recovers the data clock (as well asdata), which is used to read data from the analog-to-digital (ADC) board2806. The FGBA board 2800 also segments the digital I and Q channels.

On the transmitter side 2700, the FPGA board 2702 realizes the specialhermite correlated QAM signal as well as the necessary control signalsto control the digital-to-analog (DAC) boards 2704 to produce analog I&Qbaseband channels for the subsequent up conversion within the directconversion quad modulator 2706. The direct conversion quad modulator2706 receives an oscillator signal from oscillator 2708.

The ADC 2806 receives the I&Q signals from the quad demodulator 2808that receives an oscillator signal from 2810.

Neither power amplifier in the transmitter nor an LNA in the receiver isused since the communication will take place over a short distance. Thefrequency band of 2.4-2.5 GHz (ISM band) is selected, but any frequencyband of interest may be utilized.

MIMO uses diversity to achieve some incremental spectral efficiency.Each of the signals from the antennas acts as an independent orthogonalchannel. With QLO, the gain in spectral efficiency comes from within thesymbol and each QLO signal acts as independent channels as they are allorthogonal to one another in any permutation. However, since QLO isimplemented at the bottom of the protocol stack (physical layer), anytechnologies at higher levels of the protocol (i.e. Transport) will workwith QLO. Therefore one can use all the conventional techniques withQLO. This includes RAKE receivers and equalizers to combat fading,cyclical prefix insertion to combat time dispersion and all othertechniques using beam forming and MIMO to increase spectral efficiencyeven further.

When considering spectral efficiency of a practical wirelesscommunication system, due to possibly different practical bandwidthdefinitions (and also not strictly bandlimited nature of actual transmitsignal), the following approach would be more appropriate.

Referring now to FIG. 29, consider the equivalent discrete time system,and obtain the Shannon capacity for that system (will be denoted by Cd).Regarding the discrete time system, for example, for conventional QAMsystems in AWGN, the system will be:

y[n]=ax[n]+w[n]

where a is a scalar representing channel gain and amplitude scaling,x[n] is the input signal (QAM symbol) with unit average energy (scalingis embedded in a), y[n] is the demodulator (matched filter) outputsymbol, and index n is the discrete time index.

The corresponding Shannon capacity is:

C _(d)=log₂(1+|a| ²/σ²)

where σ2 is the noise variance (in complex dimension) and |a|2/σ2 is theSNR of the discrete time system.

Second, compute the bandwidth W based on the adopted bandwidthdefinition (e.g., bandwidth defined by −40 dBc out of band power). Ifthe symbol duration corresponding to a sample in discrete time (or thetime required to transmit C_(d) bits) is T, then the spectral efficiencycan be obtained as:

C/W=C _(d)(TW)bps/Hz

In discrete time system in AWGN channels, using Turbo or similar codeswill give performance quite close to Shannon limit C_(d). Thisperformance in discrete time domain will be the same regardless of thepulse shape used. For example, using either SRRC (square root raisedcosine) pulse or a rectangle pulse gives the same C_(d) (or C_(d)/T).However, when we consider continuous time practical systems, thebandwidths of SRRC and the rectangle pulse will be different. For atypical practical bandwidth definition, the bandwidth for a SRRC pulsewill be smaller than that for the rectangle pulse and hence SRRC willgive better spectral efficiency. In other words, in discrete time systemin AWGN channels, there is little room for improvement. However, incontinuous time practical systems, there can be significant room forimprovement in spectral efficiency.

Referring now to FIG. 30, there is illustrated a PSD plot (BLANK) ofMLO, modified MLO (MMLO) and square root raised cosine (SRRC). From theillustration in FIG. 30, demonstrates the better localization propertyof MLO. An advantage of MLO is the bandwidth. FIG. 30 also illustratesthe interferences to adjacent channels will be much smaller for MLO.This will provide additional advantages in managing, allocating orpackaging spectral resources of several channels and systems, andfurther improvement in overall spectral efficiency. If the bandwidth isdefined by the −40 dBc out of band power, the within-bandwidth PSDs ofMLO and SRRC are illustrated in FIG. 31. The ratio of the bandwidths isabout 1.536. Thus, there is significant room for improvement in spectralefficiency.

Modified MLO systems are based on block-processing wherein each blockcontains N MLO symbols and each MLO symbol has L layers. MMLO can beconverted into parallel (virtual) orthogonal channels with differentchannel SNRs as illustrated in FIG. 32. The outputs provide equivalentdiscrete time parallel orthogonal channels of MMLO.

Note that the intersymbol interference caused pulse overlapping of MLOhas been addressed by the parallel orthogonal channel conversion. As anexample, the power gain of a parallel orthogonal virtual channel of MMLOwith three layers and 40 symbols per block is illustrated in FIG. 33.FIG. 33 illustrates the channel power gain of the parallel orthogonalchannels of MMLO with three layers and T_(sim)=3. By applying a waterfilling solution, an optimal power distribution across the orthogonalchannels for a fixed transmit power may be obtained. The transmit poweron the k^(th) orthogonal channel is denoted by P_(k). Then the discretetime capacity of the MMLO can be given by:

$C_{d} = {\sum\limits_{k = 1}^{k}{\log_{2}\mspace{11mu} \left( {1 + \frac{P_{k}{a_{k}}^{2}}{\sigma_{k}^{2}}} \right)\mspace{14mu} {bits}\mspace{14mu} {per}\mspace{14mu} {block}}}$

Note that K depends on the number of MLO layers, the number of MLOsymbols per block, and MLO symbol duration.

For MLO pulse duration defined by [−t₁, t₁], and symbol durationT_(mlo), the MMLO block length is:

T _(block)=(N−1)T+2t ₁

Suppose the bandwidth of MMLO signal based on the adopted bandwidthdefinition (ACLR, OBP, or other) is W_(mmlo), then the practicalspectral efficiency of MMLO is given by:

$\frac{C_{d}}{W_{mmlo}\mspace{14mu} T_{block}} = {\frac{1}{W_{mmlo}\left\{ {{\left( {N - 1} \right)\mspace{11mu} T_{mlo}} + {2\mspace{11mu} t_{1}}} \right\}}{\sum\limits_{k = 1}^{K}{{\log_{2}\left( {1 + \frac{P_{k}{a_{k}}^{2}}{\sigma_{k}^{2}}} \right)}\mspace{14mu} \frac{bps}{Hz}}}}$

FIGS. 34-35 show the spectral efficiency comparison of MMLO with N=40symbols per block, L=3 layers, T_(mlo)=3, t₁=8, and SRRC with duration[−8T, 8T], T=1, and the roll-off factor β=0.22, at SNR of 5 dB. Twobandwidth definitions based on ACLR1 (first adjacent channel leakagepower ratio) and OBP (out of band power) are used.

FIGS. 36-37 show the spectral efficiency comparison of MMLO with L=4layers. The spectral efficiencies and the gains of MMLO for specificbandwidth definitions are shown in the following tables.

TABLE D Spectral Efficiency (bps/Hz) Gain with based on ACLR1 ≦30 dBcreference per bandwidth to SRRC SRRC 1.7859 1 MMLO (3 layers, Tmlo = 3)2.7928 1.5638 MMLO (4 layers, Tmlo = 3) 3.0849 1.7274

TABLE E Spectral Efficiency (bps/Hz) based Gain with on OBP ≦−40 dBcreference to SRRC SRRC 1.7046 1 MMLO (3 layers, Tmlo = 3) 2.3030 1.3510MMLO (4 layers, Tmlo = 3) 2.6697 1.5662

Referring now to FIGS. 38 and 39, there are provided basic blockdiagrams of low-pass-equivalent MMLO transmitters (FIG. 38) andreceivers (FIG. 39). The low-pass-equivalent MMLO transmitter 3800receives a number of input signals 3802 at a block-based transmitterprocessing 3804. The transmitter processing outputs signals to theSH(L−1) blocks 3806 which produce the I&Q outputs. These signals arethen all combined together at a combining circuit 3808 for transmission.

Within the baseband receiver (FIG. 39) 3900, the received signal isseparated and applied to a series of match filters 3902. The outputs ofthe match filters are then provided to the block-based receiverprocessing block 3904 to generate the various output streams.

Consider a block of N MLO-symbols with each MLO symbol carrying Lsymbols from L layers. Then there are NL symbols in a block. Define c(m,n)=symbol transmitted by the m-th MLO layer at the n-th MLO symbol.Write all NL symbols of a block as a column vector as follows:c=[c(0,0), c(1,0), . . . , c(L−1, 0), c(0,1), c(1,1), . . . , c(L−1, 1),. . . , c(L−1, N−1)]T. Then the outputs of the receiver matched filtersfor that transmitted block in an AWGN channel, defined by the columnvector y of length NL, can be given as y=H c+n, where H is an NL×NLmatrix representing the equivalent MLO channel, and n is a correlatedGaussian noise vector.

By applying SVD to H, we have H=U D VH where D is a diagonal matrixcontaining singular values. Transmitter side processing using V and thereceiver side processing UH, provides an equivalent system with NLparallel orthogonal channels, (i.e., y=H Vc+n and UH y=Dc+n). Theseparallel channel gains are given by diagonal elements of D. The channelSNR of these parallel channels can be computed. Note that by thetransmit and receive block-based processing, we obtain parallelorthogonal channels and hence the ISI issue has be resolved.

Since the channel SNRs of these parallel channels are not the same, wecan apply the optimal Water filling solution to compute the transmitpower on each channel given a fixed total transmit power. Using thistransmit power and corresponding channel SNR, we can compute capacity ofthe equivalent system as given in the previous report.

Issues of Fading, Muitipath, and Multi-Cell Interference

Techniques used to counteract channel fading (e.g., diversitytechniques) in conventional systems can also be applied in MMLO. Forslowly-varying multi-path dispersive channels, if the channel impulseresponse can be fed back, it can be incorporated into the equivalentsystem mentioned above, by which the channel induced ISI and theintentionally introduced MMLO ISI can be addressed jointly. For fasttime-varying channels or when channel feedback is impossible, channelequalization needs to be performed at the receiver. A block-basedfrequency-domain equalization can be applied and an oversampling wouldbe required.

If we consider the same adjacent channel power leakage for MMLO and theconventional system, then the adjacent cells' interference power wouldbe approximately the same for both systems. If interference cancellationtechniques are necessary, they can also be developed for MMLO.

Scope and System Description

This report presents the symbol error probability (or symbol error rate)performance of MLO signals in additive white Gaussian noise channel withvarious inter-symbol interference levels. As a reference, theperformance of the conventional QAM without ISI is also included. Thesame QAM size is considered for all layers of MLO and the conventionalQAM.

The MLO signals are generated from the Physicist's special Hermitefunctions:

${f_{n}\left( {t,\alpha} \right)} = {\sqrt{\frac{\alpha}{\sqrt{\pi}{n!}2^{n}}}{H_{n}\left( {\alpha \; t} \right)}^{- \frac{\alpha^{2}t^{2}}{2}}}$

where Hn(αt) is the n^(th) order Hermite polynomial. Note that thefunctions used in the lab setup correspond to

$\alpha = \frac{1}{\sqrt{2}}$

and, for consistency,

$\alpha = \frac{1}{\sqrt{2}}$

is used in this report.

MLO signals with 3, 4 or 10 layers corresponding to n=0˜2, 0˜3, or 0˜9are used and the pulse duration (the range oft) is [−8, 8] in the abovefunction.

AWGN channel with perfect synchronization is considered.

The receiver consists of matched filters and conventional detectorswithout any interference cancellation, i.e., QAM slicing at the matchedfilter outputs.

${\% \mspace{14mu} {pulse}\text{-}{overlapping}} = {\frac{T_{p} - T_{sym}}{T_{p}} \times 100\%}$

where Tp is the pulse duration (16 in the considered setup) and Tsym isthe reciprocal of the symbol rate in each MLO layer. The consideredcases are listed in the following table.

TABLE F % of Pulse Overlapping T_(sym) T_(p)    0% 16 16  12.5% 14 1618.75% 13 16   25% 12 16  37.5% 10 16 43.75% 9 16   50% 8 16 56.25% 7 16 62.5% 6 16   75% 4 16

Derivation of the Signals Used in Modulation

To do that, it would be convenient to express signal amplitude s(t) in acomplex form close to quantum mechanical formalism. Therefore thecomplex signal can be represented as:

ψ(t) = s(t) + jσ(t) where  s(t) ≡ real  signalσ(t) = imaginary  signal  (quadrature)${\sigma (t)} = {\frac{1}{\pi}{\int_{- \infty}^{\infty}{{s(\tau)}\frac{\tau}{\tau - t}}}}$${s(t)} = {{- \frac{1}{\pi}}{\int_{- \infty}^{\infty}{{\sigma (t)}\frac{\tau}{\tau - t}}}}$

Where s(t) and σ(t) are Hilbert transforms of one another and since σ(t)is quadratures of s(t), they have similar spectral components. That isif they were the amplitudes of sound waves, the ear could notdistinguish one form from the other.

Let us also define the Fourier transform pairs as follows:

${\psi (t)} = {\frac{1}{\pi}{\int_{- \infty}^{\infty}{{\phi (f)}^{j\; \omega \; t}{f}}}}$${\phi (f)} = {\frac{1}{\pi}{\int_{- \infty}^{\infty}{{\psi (t)}^{{- j}\; \omega \; t}{t}}}}$ψ^(*)(t)ψ(t) = [s(t)]² + [σ(t)]² + … ≡ signal  power

Let's also normalize all moments to M₀:

M₀ = ∫₀^(τ)s(t)  t M₀ = ∫₀^(τ)ϕ^(*)ϕ  f

Then the moments are as follows:

M₀ = ∫₀^(τ)s(t)  t M₁ = ∫₀^(τ)t  s(t)  tM₂ = ∫₀^(τ)t²  s(t)  tM_(N − 1) = ∫₀^(τ)t^(N − 1)  s(t)  t

In general, one can consider the signal s(t) be represented by apolynomial of order N, to fit closely to s(t) and use the coefficient ofthe polynomial as representation of data. This is equivalent tospecifying the polynomial in such a way that its first N “moments” M_(i)shall represent the data. That is, instead of the coefficient of thepolynomial, we can use the moments. Another method is to expand thesignal s(t) in terms of a set of N orthogonal functions φ_(k)(t),instead of powers of time. Here, we can consider the data to be thecoefficients of the orthogonal expansion. One class of such orthogonalfunctions are sine and cosine functions (like in Fourier series).

Therefore we can now represent the above moments using the orthogonalfunction with the following moments:

$\overset{\_}{t} = \frac{\int{{\psi^{*}(t)}t\mspace{11mu} {\psi (t)}\mspace{11mu} {t}}}{\int{{\psi^{*}(t)}\mspace{11mu} {\psi (t)}\mspace{11mu} {t}}}$$t^{2} = \frac{\int{{\psi^{*}(t)}t^{2}\mspace{11mu} {\psi (t)}\mspace{11mu} {t}}}{\int{{\psi^{*}(t)}\mspace{11mu} {\psi (t)}\mspace{11mu} {t}}}$$t^{n} = \frac{\int{{\psi^{*}(t)}t^{n}\mspace{11mu} {\psi (t)}\mspace{11mu} {t}}}{\int{{\psi^{*}(t)}\mspace{11mu} {\psi (t)}\mspace{11mu} {t}}}$

Similarly,

$\overset{\_}{f} = \frac{\int{{\phi^{*}(f)}f\mspace{11mu} {\phi (f)}\mspace{11mu} {f}}}{\int{{\phi^{*}(f)}\mspace{11mu} {\phi (f)}\mspace{11mu} {f}}}$$f^{2} = \frac{\int{{\phi^{*}(f)}f^{2}\mspace{11mu} {\phi (f)}\mspace{11mu} {f}}}{\int{{\phi^{*}(f)}\mspace{11mu} {\phi (f)}\mspace{11mu} {f}}}$$f^{n} = \frac{\int{{\phi^{*}(f)}f^{n}\mspace{11mu} {\phi (f)}\mspace{11mu} {f}}}{\int{{\phi^{*}(f)}\mspace{11mu} {\phi (f)}\mspace{11mu} {f}}}$

If we did not use complex signal, then:

f=0

To represent the mean values from time to frequency domains, replace:

ϕ(f)− > ψ(t) $f->{\frac{1}{2{\pi j}}\frac{\;}{t}}$

These are equivalent to somewhat mysterious rule in quantum mechanicswhere classical momentum becomes an operator:

$P_{x}->{\frac{h}{2\pi \; j}\frac{\partial\;}{\partial x}}$

Therefore using the above substitutions, we have:

${\overset{\_}{f} = {\frac{\int{{\phi^{*}(f)}f\; \phi {f}}}{\int{{\phi^{*}(f)}{\phi (f)}{f}}} = {\frac{\int{{\psi^{*}(t)}\left( \frac{1}{2{\pi j}} \right)\frac{{\psi (t)}}{t}{t}}}{\int{{\psi^{*}(t)}{\psi (t)}{t}}} = {\left( \frac{1}{2\pi \; j} \right)\frac{\int{\psi^{*}\frac{\psi}{t}{t}}}{\int{\psi^{*}\psi {t}}}}}}}\mspace{14mu}$${{And}\text{:}\mspace{14mu} {\overset{¨}{f}}^{2}} = {\frac{\int{{\phi^{*}(f)}f^{2}{\phi (f)}{f}}}{\int{{\phi^{*}(f)}{\phi (f)}{f}}} = {\frac{\int{{\psi^{*}\left( \frac{1}{{2\pi \; j}\;} \right)}^{2}\frac{^{2}}{t^{2}}\psi {t}}}{\int{\psi^{*}\psi {t}}} = {{- \left( \frac{1}{2\pi} \right)^{2}}\frac{\int{\psi^{*}\frac{^{2}}{t^{2}}\psi {t}}}{\int{\psi^{*}\psi {t}}}}}}$$\mspace{79mu} {{\overset{¨}{t}}^{2} = \frac{\int{\psi^{*}t^{2}\psi {t}}}{\int{\psi^{*}\psi {t}}}}$

We can now define an effective duration and effective bandwidth as:

${\Delta \; t} = {\sqrt{2\pi \overset{\_}{\; \left( {t - \overset{\_}{t}} \right)^{2}}} = {2{\pi \cdot {rms}}\mspace{14mu} {in}\mspace{14mu} {time}}}$${\Delta \; f} = {\sqrt{2\pi \overset{\_}{\; \left( {f - \overset{\_}{f}} \right)^{2}}} = {2{\pi \cdot {rms}}\mspace{14mu} {in}\mspace{14mu} {frequency}}}$

But we know that:

(t−{overscore (t)})² = t ² −( t )²

(f−{overscore (f)})² = f ² −( f )²

We can simplify if we make the following substitutions:

τ=t− t

Ψ(τ)=ψ(t)e ^(−j ωτ)

ω₀= ω=2π f=2πf ₀

We also know that:

(Δt)²(Δf)²=(ΔtΔf)²

And therefore:

${\left( {\Delta \; t\; \Delta \; f} \right)^{2} = {{\frac{1}{4}\left\lbrack {4\frac{\int{{\Psi^{*}(\tau)}\tau^{2}{\Psi (\tau)}{t}{\int{\frac{\Psi^{*}}{\tau}\frac{\Psi}{\tau}{\tau}}}}}{\left( {\int{{\Psi^{*}(\tau)}{\psi (\tau)}{t}}} \right)^{2}}} \right\rbrack} \geq \left( \frac{1}{4} \right)}}\;$$\left( {\Delta \; t\; \Delta \; f} \right) \geq \left( \frac{1}{2} \right)$

Now instead of

$\left( {\Delta \; t\; \Delta \; f} \right) \geq \left( \frac{1}{2} \right)$

we are interested to force the equality

$\left( {\Delta \; t\; \Delta \; f} \right) = \left( \frac{1}{2} \right)$

and see what signals satisfy the equality. Given the fixed bandwidth Δf,the most efficient transmission is one that minimizes the time-bandwidthproduct

$\left( {\Delta \; t\; \Delta \; f} \right) = \left( \frac{1}{2} \right)$

For a given bandwidth Δf, the signal that minimizes the transmission inminimum time will be a Gaussian envelope. However, we are often givennot the effective bandwidth, but always the total bandwidth f₂−f₁. Now,what is the signal shape which can be transmitted through this channelin the shortest effective time and what is the effective duration?

${\Delta \; t} = \left. \frac{\frac{1}{\left( {2\pi} \right)^{2}}{\int_{f_{1}}^{f_{2}}{\frac{\phi^{*}}{f}\ \frac{\phi}{f}}}}{\int_{f_{1}}^{f_{2}}{\phi^{*}\phi \ {f}}}\rightarrow\min \right.$

Where φ(f) is zero outside the range f₂−f₁.

To do the minimization, we would use the calculus of variations(Lagrange's Multiplier technique). Note that the denominator is constantand therefore we only need to minimize the numerator as:

$\mspace{20mu} {\left. {\Delta \; t}\rightarrow\left. \min\rightarrow{\delta {\int_{f_{1}}^{f_{2}}{\left( {{\frac{\phi^{*}}{f}\ \frac{\phi}{f}} + {{\Lambda\phi}^{*}\phi}} \right){f}}}} \right. \right. = 0}$  First  Trem${\delta {\int_{f_{1}}^{f_{2}}{\frac{\phi^{*}}{f}\frac{\phi}{f}{f}}}} = {{\int{\left( {{\frac{\phi^{*}}{f}\delta \frac{\phi}{f}} + {\frac{\phi}{f}\delta \frac{\phi^{*}}{f}}} \right){f}}} = {{\int{\left( {{\frac{\phi^{*}}{f}\frac{{\delta\phi}}{f}} + {\frac{\phi}{f}\frac{{\delta\phi}^{*}}{f}}} \right){f}}} = {{\left\lbrack {{\frac{\phi^{*}}{f}{\delta\phi}} + {\frac{\phi}{f}{\delta\phi}^{*}}} \right\rbrack_{f_{1}}^{f_{2}} - {\int{\left( {{\frac{^{2}\phi^{*}}{f^{2}}{\delta\phi}} + {\frac{^{2}\phi}{f^{2}}{\delta\phi}^{*}}} \right){f}}}} = {\int{\left( {{\frac{^{2}\phi^{*}}{f^{2}}{\delta\phi}} + {\frac{^{2}\phi}{f^{2}}{\delta\phi}^{*}}} \right){f}}}}}}$     Second  Trem     δ∫_(f₁)^(f₂)(Λϕ^(*)ϕ)f = Λ∫_(f₁)^(f₂)(ϕ^(*)δϕ + ϕδϕ^(*))f$\mspace{20mu} {{{Both}\mspace{11mu} {Trems}}\mspace{20mu} = {{\int{\left\lbrack {{\left( {\frac{^{2}\phi^{*}}{f^{2}} + {\Lambda\phi}^{*}} \right){\delta\phi}} + {\left( {\frac{^{2}\phi}{f^{2}} + {\Lambda\phi}} \right){\delta\phi}^{*}}} \right\rbrack {f}}} = 0}}$

This is only possible if and only if:

$\left( {\frac{^{2}\phi}{f^{2}} + {\Lambda\phi}} \right) = 0$

The solution to this is of the form

${\phi (f)} = {\sin \; k\; {\pi \left( \frac{f - f_{1}}{f_{2} - f_{1}} \right)}}$

Now if we require that the wave vanishes at infinity, but still satisfythe minimum time-bandwidth product:

$\left( {\Delta \; t\; \Delta \; f} \right) = \left( \frac{1}{2} \right)$

Then we have the wave equation of a Harmonic Oscillator:

${\frac{^{2}{\Psi (\tau)}}{\tau^{2}} + {\left( {\lambda - {\alpha^{2}\tau^{2}}} \right){\Psi (\tau)}}} = 0$

which vanishes at infinity only if

λ = α(2n + 1)$\psi_{n} = {{^{{- \frac{1}{2}}\omega^{2}\tau^{2}}\frac{^{n}}{\tau^{n}}^{{- \alpha^{2}}\tau^{2}}} \propto \; {H_{n}(\tau)}}$

Where H_(n)(τ) is the Hermit functions and:

$\left( {\Delta \; t\; \Delta \; f} \right) = {\frac{1}{2}\left( {{2n} + 1} \right)}$

So Hermit functions H_(n)(τ) occupy information blocks of ½, 3/2, 5/2, .. . with ½ as the minimum information quanta.

Squeezed States

Here we would derive the complete Eigen functions in the mostgeneralized form using quantum mechanical approach of Dirac algebra. Westart by defining the following operators:

$b = {\sqrt{\frac{m\; \omega^{\prime}}{2\hslash}}\left( {x + \frac{ip}{m\; \omega^{\prime}}} \right)}$$b^{+} = {{\sqrt{\frac{m\; \omega^{\prime}}{2\hslash}}{\left( {x - \frac{ip}{m\; \omega^{\prime}}} \right)\left\lbrack {b,b^{+}} \right\rbrack}} = {{1a} = {{{\lambda \; b} - {\mu \; b^{+}a^{+}}} = {{\lambda \; b^{+}} - {\mu \; b}}}}}$

Now we are ready to define Δx and Δp as:

$\left( {\Delta \; x} \right)^{2} = {{\frac{\hslash}{2m\; \omega}\left( \frac{\omega}{\omega^{\prime}} \right)} = {\frac{\hslash}{2m\; \omega}\left( {\lambda - \mu} \right)^{2}}}$$\left( {\Delta \; p} \right)^{2} = {{\frac{\hslash \; m\; \omega}{2}\left( \frac{\omega^{\prime}}{\omega} \right)} = {\frac{\hslash \; m\; \omega}{2}\left( {\lambda + \mu} \right)^{2}}}$${\left( {\Delta \; x} \right)^{2}\left( {\Delta \; p} \right)^{2}} = {\frac{\hslash^{2}}{4}\left( {\lambda^{2} - \mu^{2}} \right)^{2}}$${\Delta \; x\; \Delta \; p} = {{\frac{\hslash}{2}\left( {\lambda^{2} - \mu^{2}} \right)} = \frac{\hslash}{2}}$

Now let parameterize differently and instead of two variables λ and μ,we would use only one variable ξ as follows:

λ=sin ξ

μ=cos h ξ

λ+μ=e ^(ξ)

λ−μ=−e ^(−ξ)

Now the Eigen states of the squeezed case are:

bβ⟩ = ββ⟩ (λ a + μ a⁺)β⟩ = ββ⟩ b = UaU⁺U = ^(ξ/2(a² − a^(*²))) U⁺(ξ)aU(ξ) = acosh ξ − a⁺sinh  ξU⁺(ξ)a⁺U(ξ) = a⁺cosh  ξ − asinh ξ

We can now consider the squeezed operator:

α, ξ⟩ = U(ξ)D(α)0⟩${D(\alpha)} = {^{\frac{- {\alpha }^{2}}{2}}^{\alpha \; a^{+}}^{{- \alpha^{*}}a}}$${\alpha\rangle} = {\sum\limits_{n = 0}^{\infty}{\frac{\alpha^{n}}{\sqrt{n!}}e{n\rangle}}}$${\alpha\rangle} = {^{\frac{- {\alpha }^{2}}{2} + {\alpha \; a^{+}}}{0\rangle}}$

For a distribution P(n) we would have:

P(n) = ⟨nβ, ξ⟩²${\langle{{\alpha {}\beta},\xi}\rangle} = {\sum\limits_{n = 0}^{\infty}{\frac{\alpha^{n}}{\sqrt{n!}}^{\frac{- {\alpha }^{2}}{2}}{\langle{{n{}\beta},\xi}\rangle}}}$$^{{2{zt}} - t^{2}} = {\sum\limits_{n = 0}^{\infty}\frac{{H_{n}(z)}t^{n}}{n!}}$

Therefore the final result is:

$\begin{matrix}{{\langle{{n{}\beta},\xi}\rangle} = {\frac{\left( {\tan \; \xi} \right)^{n/2}}{2^{n/2}\left( {{n!}\cosh \; \xi} \right)^{2}}^{{{- 1}/2}{({{\beta }^{2} - {\beta^{2}\tanh \; \xi}})}}{H_{n}\left( \frac{\beta}{2\sinh \; {\xi cosh\xi}} \right)}}}\end{matrix},$

Optical Fiber Communications

The use of orbital angular momentum and multiple layer overlaymodulation processing techniques within an optical communicationsinterface environment as described with respect to FIG. 3 can provide anumber of opportunities within the optical communications environmentfor enabling the use of the greater signal bandwidths provided by theuse of optical orbital angular momentum processing, or multiple layeroverlay modulation techniques alone. FIG. 40 illustrates the generalconfiguration of an optical fiber communication system. The opticalfiber communication system 4000 includes an optical transmitter 4002 andan optical receiver 4004. The transmitter 4002 and receiver 4004communicate over an optical fiber 4006. The transmitter 4002 includesinformation within a light wavelength or wavelengths that is propagatedover the optical fiber 4006 to the optical receiver 4004.

Optical communications network traffic has been steadily increasing by afactor of 100 every decade. The capacity of single mode optical fibershas increased 10,000 times within the last three decades. Historically,the growth in the bandwidth of optical fiber communications has beensustained by information multiplexing techniques using wavelength,amplitude, phase, and polarization of light as a means for encodinginformation. Several major discoveries within the fiber-optics domainhave enabled today's optical networks. An additional discovery was ledby Charles M. Kao's groundbreaking work that recognized glass impuritieswithin an optical fiber as a major signal loss mechanism. Existing glasslosses at the time of his discovery were approximately 200 dB perkilometer at 1 micrometer.

These discoveries gave birth to optical fibers and led to the firstcommercial optical fibers in the 1970s, having an attenuation low enoughfor communication purposes in the range of approximately 20 dBs perkilometer. Referring now to FIGS. 41A-41C, there is more particularlyillustrated the single mode fiber 4102, multicore fibers 4108, andmultimode fibers 4110 described herein above. The multicore fibers 4108consist of multiple cores 4112 included within the cladding 4113 of thefiber. As can be seen in FIG. 41B, there are illustrated a 3 core fiber,7 core fiber, and 19 core fiber. Multimode fibers 4110 comprisemultimode fibers comprising a few mode fiber 4120 and a multimode fiber4122. Finally, there is illustrated a hollow core fiber 4115 including ahollow core 4114 within the center of the cladding 4116 and sheathing4118. The development of single mode fibers (SMF) such as thatillustrated at 4102 (FIG. 41A) in the early 1980s reduced pulsedispersion and led to the first fiber-optic based trans-Atlantictelephone cable. This single mode fiber included a single transmissioncore 4104 within an outer sheathing 4106. Development of indium galliumarsenide photodiodes in the early 1990s shifted the focus tonear-infrared wavelengths (1550 NM), were silica had the lowest loss,enabling extended reach of the optical fibers. At roughly the same time,the invention of erbium-doped fiber amplifiers resulted in one of thebiggest leaps in fiber capacity within the history of communication, athousand fold increase in capacity occurred over a 10 year period. Thedevelopment was mainly due to the removed need for expensive repeatersfor signal regeneration, as well as efficient amplification of manywavelengths at the same time, enabling wave division multiplexing (WDM).

Throughout the 2000s, increases in bandwidth capacity came mainly fromintroduction of complex signal modulation formats and coherentdetection, allowing information encoding using the phase of light. Morerecently, polarization division multiplexing (PDM) doubled channelcapacity. Through fiber communication based on SMFs featured tremendousgrowth in the last three decades, recent research has indicated SMFlimitations. Non-linear effects in silica play a significant role inlong range transmission, mainly through the Kerr effect, where apresence of a channel at one wavelength can change the refractive indexof a fiber, causing distortions of other wavelength channels. Morerecently, a spectral efficiency (SE) or bandwidth efficiency, referringto the transmitted information rate over a given bandwidth, has becometheoretically analyzed assuming nonlinear effects in a noisy fiberchannel. This research indicates a specific spectral efficiency limitthat a fiber of a certain length can reach for any signal to noise(SNR). Recently achieved spectral efficiency results indeed show thatthe proximity to the spectral efficiency limit, indicating the need fornew technologies to address the capacity issue in the future.

Among several possible directions for optical communications in thefuture, the introduction of new optical fibers 4006 other than singlemode fibers 4102 has shown promising results. In particular, researchershave focused on spatial dimensions in new fibers, leading to so-calledspace division multiplexing (SDM) where information is transmitted usingcores of multi-core fibers (MCF) 4108 (FIG. 41B) or mode divisionmultiplexing (MDM) or information is transmitted using modes ofmultimode fibers (MMFs) 4110 (FIG. 41C). The latest results showspectral efficiency of 91 bits/S/Hz using 12 core multicore fiber 4108for 52 kilometer long fibers and 12 bits/S/Hz using 6 mode multimodefiber 4110 and 112 kilometer long fibers. Somewhat unconventionaltransmissions at 2.08 micrometers have also been demonstrated in two 90meter long photonic crystal fibers, though these fibers had high lossesof 4.5 decibels per kilometer.

While offering promising results, these new types of fibers have theirown limitations. Being noncircularly symmetric structures, multicorefibers are known to require more complex, expensive manufacturing. Onthe other hand, multimode fibers 4110 are easily created using existingtechnologies. However, conventional multimode fibers 4110 are known tosuffer from mode coupling caused by both random perturbations in thefibers and in modal multiplexers/demultiplexers.

Several techniques have been used for mitigating mode coupling. In astrong coupling regime, modal cross talk can be compensated usingcomputationally intensive multi-input multi-output (MIMO) digital signalprocessing (DSP). While MIMO DSP leverages the technique's currentsuccess in wireless networks, the wireless network data rates areseveral orders of magnitude lower than the ones required for opticalnetworks. Furthermore, MIMO DSP complexity inevitably increases with anincreasing number of modes and no MIMO based data transmissiondemonstrations have been demonstrated in real time thus far.Furthermore, unlike wireless communication systems, optical systems arefurther complicated because of fiber's nonlinear effects. In a weakcoupling regime, where cross talk is smaller, methods that also usecomputationally intensive adapted optics, feedback algorithms have beendemonstrated. These methods reverse the effects of mode coupling bysending a desired superposition of modes at the input, so that desiredoutput modes can be obtained. This approach is limited, however, sincemode coupling is a random process that can change on the order of amillisecond in conventional fibers.

Thus, the adaptation of multimode fibers 4110 can be problematic in longhaul systems where the round trip signal propagation delay can be tensof milliseconds. Though 2×56 GB/S transmission at 8 kilometers lengthhas been demonstrated in the case of two higher order modes, none of theadaptive optics MDM methods to date have demonstrated for more than twomodes. Optical fibers act as wave guides for the information carryinglight signals that are transmitted over the fiber. Within an ideal case,optical fibers are 2D, cylindrical wave guides comprising one or severalcores surrounded by a cladding having a slightly lower refractive indexas illustrated in FIGS. 41A-41D. A fiber mode is a solution (aneigenstate) of a wave guide equation describing the field distributionthat propagates within a fiber without changing except for the scalingfactor. All fibers have a limit on the number of modes that they canpropagate, and have both spatial and polarization degrees of freedom.

Single mode fibers (SMFs) 4102 is illustrated in FIG. 41A supportpropagation of two orthogonal polarizations of the fundamental mode only(N=2). For sufficiently large core radius and/or the core claddingdifference, a fiber is multimoded for N>2 as illustrated in FIG. 41C.For optical signals having orbital angular momentums and multilayermodulation schemes applied thereto, multimode fibers 4110 that areweakly guided may be used. Weakly guided fibers have a core claddingrefractive index difference that is very small. Most glass fibersmanufactured today are weakly guided, with the exception of somephotonic crystal fibers and air-core fibers. Fiber guide modes ofmultimode fibers 4110 may be associated in step indexed groups where,within each group, modes typically having similar effective indexes aregrouped together. Within a group, the modes are degenerate. However,these degeneracies can be broken in a certain fiber profile design.

We start by describing translationally invariant waveguide withrefractive index n=n(x, y), with n_(co) being maximum refractive index(“core” of a waveguide), and n_(cl) being refractive index of theuniform cladding, and ρ represents the maximum radius of the refractiveindex n. Due to translational invariance the solutions (or modes) forthis waveguide can be written as:

E _(j)(x,y,z)=e _(j)(x,y)e ^(iβ) ^(j) ^(z),

H _(j)(x,y,z)=h _(j)(x,y)e ^(iβ) ^(j) ^(z),

where β_(j) is the propagation constant of the j-th mode. Vector waveequation for source free Maxwell's equation can be written in this caseas:

(∇² +n ² k ²−β_(j) ²)e _(j)=−(∇_(t) +iβ _(j) Z)(e _(tj)·∇_(t) ln(n ²))

(∇² +n ² k ²β_(j) ²)h _(j)−(∇_(t) ln(n ²))×(

(∇

_(t) +iβ _(j){circumflex over ({circumflex over (z)})×h _(j))

where k=2π/λ is the free-space wavenumber, λ is a free-space wavelength,e_(t)=e_(x){circumflex over (x)}+e_(y)ŷ is a transverse part of theelectric field, ∇² is a transverse Laplacian and ∇_(t) transverse vectorgradient operator. Waveguide polarization properties are built into thewave equation through the ∇_(t) In(n²) terms and ignoring them wouldlead to the scalar wave equation, with linearly polarized modes. Whileprevious equations satisfy arbitrary waveguide profile n(x, y), in mostcases of interest, profile height parameter Δ can be considered smallΔ<<1, in which case waveguide is said to be weakly guided, or thatweakly guided approximation (WGA) holds. If this is the case, aperturbation theory can be applied to approximate the solutions as:

E(x,y,z)=e(x,y)e ^(i(β+β)z)=(e _(t) +{circumflex over (z)}e _(z))e^(i(β+β)z)

H(x,y,z)=h(x,y)e ^(i(β+β)z)=(h _(t) +{circumflex over (z)}h _(z))e^(i(β+β)z)

where subscripts t and z denote transverse and longitudinal componentsrespectively. Longitudinal components can be considered much smaller inWGA and we can approximate (but not neglect) them as:

$e_{z} = {\frac{{\left( {2\Delta} \right)}^{\frac{1}{2}}}{V}\left( {\rho {\nabla_{t}{\cdot e_{t}}}} \right)}$$h_{z\;} = {\frac{{\left( {2\Delta} \right)}^{\frac{1}{2}}}{V}\left( {\rho \; {\nabla_{t}{\cdot h_{t}}}} \right)}$

Where Δ and ∇ are profile height and fiber parameters and transversalcomponents satisfy the simplified wave equation.

(∇z+n ² k ²−β_(j) ^(z))e _(j)=0

Though WGA simplified the waveguide equation, further simplification canbe obtained by assuming circularly symmetric waveguide (such as idealfiber). If this is the case refractive index that can be written as:

n(r)=n ² _(co)(1−2f(R)Δ)

where f(R)≧0 is a small arbitrary profile variation.

For a circularly symmetric waveguide, we would have propagationconstants β_(lm) that are classified using azimuthal (l) and radial (m)numbers. Another classification uses effective indices n (sometimesnoted as n^(eff) _(lm) or simply n_(eff), that are related topropagation constant as: β_(lm)=kn^(eff)). For the case of l=0, thesolutions can be separated into two classes that have either transverseelectric (TE_(0m)) or transverse magnetic (TM_(0m)) fields (calledmeridional modes). In the case of l≠0, both electric and magnetic fieldhave z-component, and depending on which one is more dominant, so-calledhybrid modes are denoted as: HE_(lm) and EH_(lm).

Polarization correction δβ has different values within the same group ofmodes with the same orbital number (l), even in the circularly symmetricfiber. This is an important observation that led to development of aspecial type of fiber.

In case of a step refractive index, solutions are the Bessel functionsof the first kind, J_(l)(r), in the core region, and modified Besselfunctions of the second kind, K_(l)(r), in the cladding region.

In the case of step-index fiber the groups of modes are almostdegenerate, also meaning that the polarization correction δβ can beconsidered very small. Unlike HE₁₁ modes, higher order modes (HOMs) canhave elaborate polarizations. In the case of circularly symmetric fiber,the odd and even modes (for example HE^(odd) and HE^(even) modes) arealways degenerate (i.e. have equal n_(eff)), regardless of the indexprofile. These modes will be non-degenerate only in the case ofcircularly asymmetric index profiles.

Referring now to FIG. 42, there are illustrated the first six modeswithin a step indexed fiber for the groups L=0 and L=1.

When orbital angular momentums are applied to the light wavelengthwithin an optical transmitter of an optical fiber communication system,the various orbital angular momentums applied to the light wavelengthmay transmit information and be determined within the fiber mode.

Angular momentum density (M) of light in a medium is defined as:

$M = {{\frac{1}{c^{2}}r \times \left( {E \times H} \right)} = {{r \times P} = {\frac{1}{c^{2}}r \times S}}}$

with r as position, E electric field, H magnetic field, P linearmomentum density and S Poynting vector.

The total angular momentum (J), and angular momentum flux (Φ_(M)) can bedefined as:

J=∫∫∫MdV

Φ_(M) =∫∫MdA

In order to verify whether certain mode has an OAM let us look at thetime averages of the angular momentum flux Φ_(M):

(Φ_(M))=17 ∫(M)dA

as well as the time average of the energy flux:

$\left( \Phi_{W} \right) = {\int{\int{\frac{\left( S_{2} \right)}{c}{A}}}}$

Because of the symmetry of radial and axial components about the fiberaxis, we note that the integration in equation will leave onlyz-component of the angular momentum density non zero. Hence:

${\langle M\rangle} = {{\langle M\rangle}_{z} = {\frac{1}{c^{2}}r \times {\langle{E \times H}\rangle}_{z}}}$

and knowing (S)=Re{S} and S=½E×H* leads to:

$S_{\varphi} = {\frac{1}{2}\left( {{{- E_{r}}H_{z}^{*}} + {E_{z}H_{r}^{*}}} \right)}$$S_{z} = {\frac{1}{2}\left( {{E_{x}H_{y}^{*}} - {E_{y}H_{x}^{*}}} \right)}$

Let us now focus on a specific linear combination of the HE_(l+1,m)^(even) and HE_(l+1,m) ^(odd) modes with π/2 phase shift among them:

V _(lm) ⁺ =HE _(l+1,m) ^(even) +iEH _(l+1,m) ^(odd)

The idea for this linear combination comes from observing azimuthaldependence of the HE_(l+1,m) ^(even) and HE_(l+1,m) ^(odd) modescomprising cos(φ) and sin (φ). If we denote the electric field ofHE_(l+1,m) ^(even) and HE_(l+1,m) ^(odd) modes as e₁ and e₂,respectively, and similarly, denote their magnetic fields as h₁ and h₂,the expression for this new mode can be written as:

e=e ₁ +ie ₂,  (2.35)

h=h ₁ +ih ₂.  (2.36)

then we derive:

e_(r) = ^(( + 1)ϕ)F_(i)(R)$h_{z} = {^{{{({ + 1})}}\phi}n_{co}*\left( \frac{\varepsilon_{0}}{\mu_{0}} \right)^{\frac{1}{2}}\frac{\left( {2\Delta} \right)^{\frac{1}{2}}}{V}G_{l}^{-}}$$e_{z} = {{}^{{{({ + 1})}}\phi}\frac{\left( {2\Delta} \right)^{\frac{1}{2}}}{V}G_{i}^{-}}$$h_{r} = {{- }\; ^{{{({ + 1})}}\phi}{n_{co}\left( \frac{\varepsilon_{0}}{\mu_{0}} \right)}^{\frac{1}{2}}{F_{l}(R)}}$

Where F₁(R) is the Bessel function and

$G_{i}^{\pm} = {\frac{F_{l}}{R} \pm {\frac{l}{R}F_{l}}}$

We note that all the quantities have e^(i(l+1)φ) dependence thatindicates these modes might have OAM, similarly to the free space case.Therefore the azimuthal and the longitudinal component of the Poyntingvector are:

$S_{\phi} = {{- {n_{co}\left( \frac{\varepsilon_{0}}{\mu_{0}} \right)}^{\frac{1}{2}}}\frac{\left( {2\Delta} \right)^{\frac{1}{2}}}{V}{Re}\left\{ {F_{l}^{*}G_{l}^{-}} \right\}}$$S_{z} = {{n_{c\; o}\left( \frac{\varepsilon_{0}}{\mu_{0}} \right)}^{\frac{1}{2}}\left\lbrack F_{l} \right\rbrack}^{2}$

The ratio of the angular momentum flux to the energy flux thereforebecomes:

$\frac{Ø_{M}}{Ø_{W}} = \frac{ + 1}{\omega}$

We note that in the free-space case, this ratio is similar:

$\frac{Ø_{M}}{Ø_{W}} = \frac{\sigma + 1}{\omega}$

where σ represents the polarization of the beam and is bounded to be−1<σ<1. In our case, it can be easily shown that SAM of the V⁺ state, is1, leading to important conclusion that the OAM of the V^(+lm) state isl. Hence, this shows that, in an ideal fiber, OAM mode exists.

Thus, since an orbital angular momentum mode may be detected within theideal fiber, it is possible to encode information using this OAM mode inorder to transmit different types of information having differentorbital angular momentums within the same optical wavelength.

The above description with respect to optical fiber assumed an idealscenario of perfectly symmetrical fibers having no longitudinal changeswithin the fiber profile. Within real world fibers, random perturbationscan induce coupling between spatial and/or polarization modes, causingpropagating fields to evolve randomly through the fiber. The randomperturbations can be divided into two classes, as illustrated in FIG.43. Within the random perturbations 4302, the first class comprisesextrinsic perturbations 4304. Extrinsic perturbations 4304 includestatic and dynamic fluctuations throughout the longitudinal direction ofthe fiber, such as the density and concentration fluctuations natural torandom glassy polymer materials that are included within fibers. Thesecond class includes extrinsic variations 4306 such as microscopicrandom bends caused by stress, diameter variations, and fiber coredefects such as microvoids, cracks, or dust particles.

Mode coupling can be described by field coupling modes which account forcomplex valued modal electric field amplitudes, or by power couplingmodes, which is a simplified description that accounts only for realvalue modal powers. Early multimode fiber systems used incoherent lightemitting diode sources and power coupling models were widely used todescribe several properties including steady state, modal powerdistributions, and fiber impulse responses. While recent multimode fibersystems use coherent sources, power coupling modes are still used todescribe effects such as reduced differential group delays and plasticmultimode fibers.

By contrast, single mode fiber systems have been using laser sources.The study of random birefringence and mode coupling in single modefibers which leads to polarization mode dispersion (PMD), uses fieldcoupling modes which predict the existence of principal states ofpolarization (PSPs). PSPs are polarization states shown to undergominimal dispersion and are used for optical compensation of polarizationmode dispersion in direct detection single mode fiber systems. In recentyears, field coupling modes have been applied to multimode fibers,predicting principal mode which are the basis for optical compensationof modal dispersion in direct detection multimode fiber systems.

Mode coupling can be classified as weak or strong, depending on whetherthe total system length of the optical fiber is comparable to, or muchlonger than, a length scale over which propagating fields remaincorrelated. Depending on the detection format, communication systems canbe divided into direct and coherent detection systems. In directdetection systems, mode coupling must either be avoided by carefuldesign of fibers and modal D (multiplexers) and/or mitigated by adaptiveoptical signal processing. In systems using coherent detection, anylinear cross talk between modes can be compensated by multiple inputmultiple output (MIMO) digital signal processing (DSP), as previouslydiscussed, but DSP complexity increases with an increasing number ofmodes.

Referring now to FIG. 44, there were illustrated the intensity patternsof the first order mode group within a vortex fiber. Arrows 4402 withinthe illustration show the polarization of the electric field within thefiber. The top row illustrates vector modes that are the exact vectorsolutions, and the bottom row shows the resultant, unstable LP11 modescommonly obtained at a fiber output. Specific linear combinations ofpairs of top row modes resulting in the variety of LP11 modes obtainedat the fiber output. Coupled mode 4402 is provided by the coupled pairof mode 4404 and 4406. Coupled mode 4404 is provided by the coupled pairof mode 4404 and mode 4408. Coupled mode 4416 is provided by the coupledpair of mode 4406 and mode 4410, and coupled mode 4418 is provided bythe coupled pair of mode 4408 and mode 4410.

Typically, index separation of two polarizations and single mode fibersis on the order of 10-7. While this small separation lowers the PMD ofthe fiber, external perturbations can easily couple one mode intoanother, and indeed in a single mode fiber, arbitrary polarizations aretypically observed at the output. Simple fiber polarization controllerthat uses stress induced birefringence can be used to achieve anydesired polarization at the output of the fiber.

By the origin, mode coupling can be classified as distributed (caused byrandom perturbations in fibers), or discrete (caused at the modalcouplers and the multiplexers). Most importantly, it has been shown thatsmall, effective index separation among higher order modes is the mainreason for mode coupling and mode instabilities. In particular, thedistributed mode coupling has been shown to be inversely proportional toΔ−P with P greater than 4, depending on coupling conditions. Modeswithin one group are degenerate. For this reason, in most multimodefiber modes that are observed in the fiber output are in fact the linearcombinations of vector modes and are linearly polarized states. Hence,optical angular momentum modes that are the linear combination of the HEeven, odd modes cannot coexist in these fibers due to coupling todegenerate TE01 and TM01 states.

Thus, the combination of the various OAM modes is not likely to generatemodal coupling within the optical systems and by increasing the numberof OAM modes, the reduction in mode coupling is further benefited.

Referring now to FIGS. 45A and 45B, there is illustrated the benefit ofeffective index separation in first order modes. FIG. 45A illustrates atypical step index multimode fiber that does not exhibit effective indexseparation causing mode coupling. The mode TM₀₁ HE^(even) ₂₁, modeHE^(odd) ₂₁, and mode TE₀₁ have little effective index separation, andthese modes would be coupled together. Mode HE^(x,1) ₁₁ has an effectiveindex separation such that this mode is not coupled with these othermodes.

This can be compared with the same modes in FIG. 45B. In this case,there is an effective separation 4502 between the TM₀₁ mode and theHE^(even) ₂₁ mode and the TE₀₁ mode and the HE^(odd) ₂₁ mode. Thiseffective separation causes no mode coupling between these mode levelsin a similar manner that was done in the same modes in FIG. 45A.

In addition to effective index separation, mode coupling also depends onthe strength of perturbation. An increase in the cladding diameter of anoptical fiber can reduce the bend induced perturbations in the fiber.Special fiber design that includes the trench region can achieveso-called bend insensitivity, which is predominant in fiber to the home.Fiber design that demonstrates reduced bends and sensitivity of higherorder Bessel modes for high power lasers have been demonstrated. Mostimportant, a special fiber design can remove the degeneracy of the firstorder mode, thus reducing the mode coupling and enabling the OAM modesto propagate within these fibers.

Topological charge may be multiplexed to the wave length for eitherlinear or circular polarization. In the case of linear polarizations,topological charge would be multiplexed on vertical and horizontalpolarization. In case of circular polarization, topological charge wouldbe multiplexed on left hand and right hand circular polarization.

The topological charges can be created using Spiral Phase Plates (SPPs)such as that illustrated in FIG. 11E, phase mask holograms or a SpatialLight Modulator (SLM) by adjusting the voltages on SLM which createsproperly varying index of refraction resulting in twisting of the beamwith a specific topological charge. Different topological charges can becreated and muxed together and de-muxed to separate charges.

As Spiral Phase plates can transform a plane wave (l=0) to a twistedwave of a specific helicity (i.e. l=+1), Quarter Wave Plates (QWP) cantransform a linear polarization (s=0) to circular polarization (i.e.s=+1).

Cross talk and multipath interference can be reduced usingMultiple-input-Multiple-Output (MIMO).

Most of the channel impairments can be detected using a control or pilotchannel and be corrected using algorithmic techniques (closed loopcontrol system)

Free Space Communications

An additional configuration in which the optical angular momentumprocessing and multi-layer overlay modulation technique described hereinabove may prove useful within the optical network framework is use withfree-space optics communications. Free-space optics systems provide anumber of advantages over traditional UHF RF based systems from improvedisolation between the systems, the size and the cost of thereceivers/transmitters, lack of RF licensing laws, and by combiningspace, lighting, and communication into the same system. Referring nowto FIG. 46, there is illustrated an example of the operation of afree-space communication system. The free-space communication systemutilizes a free-space optics transmitter 4602 that transmits a lightbeam 4604 to a free-space optics receiver 4606. The major differencebetween a fiber-optic network and a free-space optic network is that theinformation beam is transmitted through free space rather than over afiber-optic cable. This causes a number of link difficulties, which willbe more fully discussed herein below. Free-space optics is a line ofsight technology that uses the invisible beams of light to provideoptical bandwidth connections that can send and receive up to 2.5 Gbpsof data, voice, and video communications between a transmitter 4602 anda receiver 4606. Free-space optics uses the same concepts asfiber-optics, except without the use of a fiber-optic cable. Free-spaceoptics systems provide the light beam 4604 within the infrared (IR)spectrum, which is at the low end of the light spectrum. Specifically,the optical signal is in the range of 300 Gigahertz to I Terahertz interms of wavelength.

Presently existing free-space optics systems can provide data rates ofup to 10 Gigabits per second at a distance of up to 2.5 kilometers. Inouter space, the communications range of free space opticalcommunications is currently on the order of several thousand kilometers,but has the potential to bridge interplanetary distances of millions ofkilometers, using optical telescopes as beam expanders. In January of2013, NASA used lasers to beam an image of the Mona Lisa to the LunarReconnaissance Orbiter roughly 240,000 miles away. To compensate foratmospheric interference, an error correction code algorithm, similar tothat used within compact discs, was implemented.

The distance records for optical communications involve detection andemission of laser light by space probes. A two-way distance record forcommunication was established by the Mercury Laser Altimeter instrumentaboard the MESSENGER spacecraft. This infrared diode neodymium laser,designed as a laser altimeter for a Mercury Orbiter mission, was able tocommunicate across a distance of roughly 15,000,000 miles (24,000,000kilometers) as the craft neared Earth on a fly by in May of 2005. Theprevious record had been set with a one-way detection of laser lightfrom Earth by the Galileo Probe as two ground based lasers were seenfrom 6,000,000 kilometers by the outbound probe in 1992. Researchersused a white LED based space lighting system for indoor local areanetwork communications.

Referring now to FIG. 47, there is illustrated a block diagram of afree-space optics system using orbital angular momentum and multileveloverlay modulation according to the present disclosure. The OAM twistedsignals, in addition to being transmitted over fiber, may also betransmitted using free optics. In this case, the transmission signalsare generated within transmission circuitry 4702 at each of the FSOtransceivers 4704. Free-space optics technology is based on theconnectivity between the FSO based optical wireless units, eachconsisting of an optical transceiver 4704 with a transmitter 4702 and areceiver 4706 to provide full duplex open pair and bidirectional closedpairing capability. Each optical wireless transceiver unit 4704additionally includes an optical source 4708 plus a lens or telescope4710 for transmitting light through the atmosphere to another lens 4710receiving the information. At this point, the receiving lens ortelescope 4710 connects to a high sensitivity receiver 4706 via opticalfiber 4712. The transmitting transceiver 4704 a and the receivingtransceiver 4704 b have to have line of sight to each other. Trees,buildings, animals, and atmospheric conditions all can hinder the lineof sight needed for this communications medium. Since line of sight isso critical, some systems make use of beam divergence or a diffused beamapproach, which involves a large field of view that toleratessubstantial line of sight interference without significant impact onoverall signal quality. The system may also be equipped with autotracking mechanism 4714 that maintains a tightly focused beam on thereceiving transceiver 3404 b, even when the transceivers are mounted ontall buildings or other structures that sway.

The modulated light source used with optical source 4708 is typically alaser or light emitting diode (LED) providing the transmitted opticalsignal that determines all the transmitter capabilities of the system.Only the detector sensitivity within the receiver 4706 plays an equallyimportant role in total system performance. For telecommunicationspurposes, only lasers that are capable of being modulated at 20 Megabitsper second to 2.5 Gigabits per second can meet current marketplacedemands. Additionally, how the device is modulated and how muchmodulated power is produced are both important to the selection of thedevice. Lasers in the 780-850 nm and 1520-1600 nm spectral bands meetfrequency requirements.

Commercially available FSO systems operate in the near IR wavelengthrange between 750 and 1600 nm, with one or two systems being developedto operate at the IR wavelength of 10,000 nm. The physics andtransmissions properties of optical energy as it travels through theatmosphere are similar throughout the visible and near IR wavelengthrange, but several factors that influence which wavelengths are chosenfor a particular system.

The atmosphere is considered to be highly transparent in the visible andnear IR wavelength. However, certain wavelengths or wavelength bands canexperience severe absorption. In the near IR wavelength, absorptionoccurs primarily in response to water particles (i.e., moisture) whichare an inherent part of the atmosphere, even under clear weatherconditions. There are several transmission windows that are nearlytransparent (i.e., have an attenuation of less than 0.2 dB perkilometer) within the 700-10,000 nm wavelength range. These wavelengthsare located around specific center wavelengths, with the majority offree-space optics systems designed to operate in the windows of 780-850nm and 1520-1600 nm.

Wavelengths in the 780-850 nm range are suitable for free-space opticsoperation and higher power laser sources may operate in this range. At780 nm, inexpensive CD lasers may be used, but the average lifespan ofthese lasers can be an issue. These issues may be addressed by runningthe lasers at a fraction of their maximum rated output power which willgreatly increase their lifespan. At around 850 nm, the optical source4708 may comprise an inexpensive, high performance transmitter anddetector components that are readily available and commonly used innetwork transmission equipment. Highly sensitive silicon (SI) avalanchephotodiodes (APD) detector technology and advanced vertical cavityemitting laser may be utilized within the optical source 4708.

VCSEL technology may be used for operation in the 780 to 850 nm range.Possible disadvantage of this technology include beam detection throughthe use of a night vision scope, although it is still not possible todemodulate a perceived light beam using this technique.

Wavelengths in the 1520-1600 nm range are well-suited for free-spacetransmission, and high quality transmitter and detector components arereadily available for use within the optical source block 4708. Thecombination of low attenuation and high component availability withinthis wavelength range makes the development of wavelength divisionmultiplexing (WDM) free-space optics systems feasible. However,components are generally more expensive and detectors are typically lesssensitive and have a smaller receive surface area when compared withsilicon avalanche photodiode detectors that operator at the 850 nmwavelength. These wavelengths are compatible with erbium-doped fiberamplifier technology, which is important for high power (greater than500 milliwatt) and high data rate (greater than 2.5 Gigabytes persecond) systems. Fifty to 65 times as much power can be transmitted atthe 1520-1600 nm wavelength than can be transmitted at the 780-850 nmwavelength for the same eye safety classification. Disadvantages ofthese wavelengths include the inability to detect a beam with a nightvision scope. The night vision scope is one technique that may be usedfor aligning the beam through the alignment circuitry 4714. Class 1lasers are safe under reasonably foreseeable operating conditionsincluding the use of optical instruments for intrabeam viewing. Class 1systems can be installed at any location without restriction.

Another potential optical source 4708 comprised Class 1M lasers. Class1M laser systems operate in the wavelength range from 302.5 to 4000 nm,which is safe under reasonably foreseeable conditions, but may behazardous if the user employs optical instruments within some portion ofthe beam path. As a result, Class 1M systems should only be installed inlocations where the unsafe use of optical aids can be prevented.Examples of various characteristics of both Class 1 and Class 1M lasersthat may be used for the optical source 4708 are illustrated in Table Gbelow.

TABLE G Laser Power Aperture Size Distance Power Density Classification(mW) (mm) (m) (mW/cm²) 850-nm Wavelength Class 1 0.78 7 14 2.03 50 20000.04 Class 1M 0.78 7 100 2.03 500 7 14 1299.88 50 2000 25.48 1550-nmWavelength Class 1 10 7 14 26.00 25 2000 2.04 Class 1M 10 3.5 100 103.99500 7 14 1299.88 25 2000 101.91

The 10,000 nm wavelength is relatively new to the commercial free spaceoptic arena and is being developed because of better fog transmissioncapabilities. There is presently considerable debate regarding thesecharacteristics because they are heavily dependent upon fog type andduration. Few components are available at the 10,000 nm wavelength, asit is normally not used within telecommunications equipment.Additionally, 10,000 nm energy does not penetrate glass, so it isill-suited to behind window deployment.

Within these wavelength windows, FSO systems should have the followingcharacteristics. The system should have the ability to operate at higherpower levels, which is important for longer distance FSO systemtransmissions. The system should have the ability to provide high speedmodulation, which is important for high speed FSO systems. The systemshould provide a small footprint and low power consumption, which isimportant for overall system design and maintenance. The system shouldhave the ability to operate over a wide temperature range without majorperformance degradations such that the systems may prove useful foroutdoor systems. Additionally, the mean time between failures shouldexceed 10 years. Presently existing FSO systems generally use VCSELS foroperation in the shorter IR wavelength range, and Fabry-Pérot ordistributed feedback lasers for operation in the longer IR wavelengthrange. Several other laser types are suitable for high performance FSOsystems.

A free-space optics system using orbital angular momentum processing andmulti-layer overlay modulation would provide a number of advantages. Thesystem would be very convenient. Free-space optics provides a wirelesssolution to a last-mile connection, or a connection between twobuildings. There is no necessity to dig or bury fiber cable. Free-spaceoptics also requires no RF license. The system is upgradable and itsopen interfaces support equipment from a variety of vendors. The systemcan be deployed behind windows, eliminating the need for costly rooftopright. It is also immune to radiofrequency interference or saturation.The system is also fairly speedy. The system provides 2.5 Gigabits persecond of data throughput. This provides ample bandwidth to transferfiles between two sites. With the growth in the size of files,free-space optics provides the necessary bandwidth to transfer thesefiles efficiently.

Free-space optics also provides a secure wireless solution. The laserbeam cannot be detected with a spectral analyzer or RF meter. The beamis invisible, which makes it difficult to find. The laser beam that isused to transmit and receive the data is very narrow. This means that itis almost impossible to intercept the data being transmitted. One wouldhave to be within the line of sight between the receiver and thetransmitter in order to be able to accomplish this feat. If this occurs,this would alert the receiving site that a connection has been lost.Thus, minimal security upgrades would be required for a free-spaceoptics system.

However, there are several weaknesses with free-space optics systems.The distance of a free-space optics system is very limited. Currentlyoperating distances are approximately within 2 kilometers. Although thisis a powerful system with great throughput, the limitation of distanceis a big deterrent for full-scale implementation. Additionally, allsystems require line of sight be maintained at all times duringtransmission. Any obstacle, be it environmental or animals can hinderthe transmission. Free-space optic technology must be designed to combatchanges in the atmosphere which can affect free-space optic systemperformance capacity.

Something that may affect a free-space optics system is fog. Dense fogis a primary challenge to the operation of free-space optics systems.Rain and snow have little effect on free-space optics technology, butfog is different. Fog is a vapor composed of water droplets which areonly a few hundred microns in diameter, but can modify lightcharacteristics or completely hinder the passage of light through acombination of absorption, scattering, and reflection. The primaryanswer to counter fog when deploying free-space optic based wirelessproducts is through a network design that shortens FSO linked distancesand adds network redundancies.

Absorption is another problem. Absorption occurs when suspended watermolecules in the terrestrial atmosphere extinguish photons. This causesa decrease in the power density (attenuation) of the free space opticsbeam and directly affects the availability of the system. Absorptionoccurs more readily at some wavelengths than others. However, the use ofappropriate power based on atmospheric conditions and the use of spatialdiversity (multiple beams within an FSO based unit), helps maintain therequired level of network availability.

Solar interference is also a problem. Free-space optics systems use ahigh sensitivity receiver in combination with a larger aperture lens. Asa result, natural background light can potentially interfere withfree-space optics signal reception. This is especially the case with thehigh levels of background radiation associated with intense sunlight. Insome instances, direct sunlight may case link outages for periods ofseveral minutes when the sun is within the receiver's field of vision.However, the times when the receiver is most susceptible to the effectsof direct solar illumination can be easily predicted. When directexposure of the equipment cannot be avoided, the narrowing of receiverfield of vision and/or using narrow bandwidth light filters can improvesystem performance. Interference caused by sunlight reflecting off of aglass surface is also possible.

Scattering issues may also affect connection availability. Scattering iscaused when the wavelength collides with the scatterer. The physicalsize of the scatterer determines the type of scattering. When thescatterer is smaller than the wavelength, this is known as Rayleighscattering. When a scatterer is of comparable size to the wavelengths,this is known as Mie scattering. When the scattering is much larger thanthe wavelength, this is known as non-selective scattering. Inscattering, unlike absorption, there is no loss of energy, only adirectional redistribution of energy that may have significant reductionin beam intensity over longer distances.

Physical obstructions such as flying birds or construction cranes canalso temporarily block a single beam free space optics system, but thistends to cause only short interruptions. Transmissions are easily andautomatically resumed when the obstacle moves. Optical wireless productsuse multibeams (spatial diversity) to address temporary abstractions aswell as other atmospheric conditions, to provide for greateravailability.

The movement of buildings can upset receiver and transmitter alignment.Free-space optics based optical wireless offerings use divergent beamsto maintain connectivity. When combined with tracking mechanisms,multiple beam FSO based systems provide even greater performance andenhanced installation simplicity.

Scintillation is caused by heated air rising from the Earth or man-madedevices such as heating ducts that create temperature variations amongdifferent pockets of air. This can cause fluctuations in signalamplitude, which leads to “image dancing” at the free-space optics basedreceiver end. The effects of this scintillation are called “refractiveturbulence.” This causes primarily two effects on the optical beams.Beam wander is caused by the turbulent eddies that are no larger thanthe beam. Beam spreading is the spread of an optical beam as itpropagates through the atmosphere.

Referring now to FIGS. 48A through 48D, in order to achieve higher datacapacity within optical links, an additional degree of freedom frommultiplexing multiple data channels must be exploited. Moreover, theability, to use two different orthogonal multiplexing techniquestogether has the potential to dramatically enhance system performanceand increased bandwidth.

One multiplexing technique which may exploit the possibilities is modedivision multiplexing (MDM) using orbital angular momentum (OAM). OAMmode refers to laser beams within a free-space optical system orfiber-optic system that have a phase term of e^(ilφ) in their wavefronts, in which φ is the azimuth angle and l determines the OAM value(topological charge). In general, OAM modes have a “donut-like” ringshaped intensity distribution. Multiple spatial collocated laser beams,which carry different OAM values, are orthogonal to each other and canbe used to transmit multiple independent data channels on the samewavelength. Consequently, the system capacity and spectral efficiency interms of bits/S/Hz can be dramatically increased. Free-spacecommunications links using OAM may support 100 Tbits/capacity. Varioustechniques for implementing this as illustrated in FIGS. 48A through 48Dinclude a combination of multiple beams 4802 having multiple differentOAM values 4804 on each wavelength. Thus, beam 4802 includes OAM values,OAM1 and OAM4. Beam 4806 includes OAM value 2 and OAM value 5. Finally,beam 4808 includes OAM3 value and OAM6 value. Referring now to FIG. 48B,there is illustrated a single beam wavelength 4810 using a first groupof OAM values 4812 having both a positive OAM value 4812 and a negativeOAM value 4814. Similarly, OAM2 value may have a positive value 4816 anda negative value 4818 on the same wavelength 4810.

FIG. 48C illustrates the use of a wavelength 4820 having polarizationmultiplexing of OAM value. The wavelength 4820 can have multiple OAMvalues 4822 multiplexed thereon. The number of available channels can befurther increased by applying left or right handed polarization to theOAM values. Finally, FIG. 48D illustrates two groups of concentric rings4860, 4862 for a wavelength having multiple OAM values.

Wavelength distribution multiplexing (WDM) has been widely used toimprove the optical communication capacity within both fiber-opticsystems and free-space communication system. OAM mode multiplexing andWDM are mutually orthogonal such that they can be combined to achieve adramatic increase in system capacity. Referring now to FIG. 49, there isillustrated a scenario where each WDM channel 4902 contains manyorthogonal OAM beam 4904. Thus, using a combination of orbital angularmomentum with wave division multiplexing, a significant enhancement incommunication link to capacity may be achieved.

Current optical communication architectures have considerable routingchallenges. A routing protocol for use with free-space optic system musttake into account the line of sight requirements for opticalcommunications within a free-space optics system. Thus, a free-spaceoptics network must be modeled as a directed hierarchical random sectorgeometric graph in which sensors route their data via multi-hop paths toa base station through a cluster head. This is a new efficient routingalgorithm for local neighborhood discovery and a base station uplink anddownlink discovery algorithm. The routing protocol requires order Olog(n) storage at each node versus order O(n) used within currenttechniques and architectures.

Current routing protocols are based on link state, distance vectors,path vectors, or source routing, and they differ from the new routingtechnique in significant manners. First, current techniques assume thata fraction of the links are bidirectional. This is not true within afree-space optic network in which all links are unidirectional. Second,many current protocols are designed for ad hoc networks in which therouting protocol is designed to support multi-hop communications betweenany pair of nodes. The goal of the sensor network is to route sensorreadings to the base station. Therefore, the dominant traffic patternsare different from those in an ad hoc network. In a sensor network, nodeto base stations, base station to nodes, and Local neighborhoodcommunication are mostly used.

Recent studies have considered the effect of unidirectional links andreport that as many as 5 percent to 10 percent of links and wireless adhoc networks are unidirectional due to various factors. Routingprotocols such as DSDV and AODV use a reverse path technique, implicitlyignoring such unidirectional links and are therefore not relevant inthis scenario. Other protocols such as DSR, ZRP, or ZRL have beendesigned or modified to accommodate unidirectionality by detectingunidirectional links and then providing bidirectional abstraction forsuch links. Referring now to FIG. 50, the simplest and most efficientsolution for dealing with unidirectionality is tunneling, in whichbidirectionality is emulated for a unidirectional link by usingbidirectional links on a reverse back channel to establish the tunnel.Tunneling also prevents implosion of acknowledgement packets and loopingby simply pressing link layer acknowledgements for tunneled packetsreceived on a unidirectional link. Tunneling, however, works well inmostly bidirectional networks with few unidirectional links.

Within a network using only unidirectional links such as a free-spaceoptical network, systems such as that illustrated in FIGS. 50 and 51would be more applicable. Nodes within a unidirectional network utilizea directional transmit 5002 transmitting from the node 5000 in a single,defined direction. Additionally, each node 5000 includes anomnidirectional receiver 5004 which can receive a signal coming to thenode in any direction. Also, as discussed here and above, the node 5000would also include a O log(n) storage 5006. Thus, each node 5000 provideonly unidirectional communications links. Thus, a series of nodes 5000as illustrated in FIG. 51 may unidirectionally communicate with anyother node 5000 and forward communication from one desk location toanother through a sequence of interconnected nodes.

Topological charge may be multiplexed to the wave length for eitherlinear or circular polarization. In the case of linear polarizations,topological charge would be multiplexed on vertical and horizontalpolarization. In case of circular polarization, topological charge wouldbe multiplexed on left hand and right hand circular polarizations.

The topological charges can be created using Spiral Phase Plates (SPPs)such as that illustrated in FIG. 11E, phase mask holograms or a SpatialLight Modulator (SLM) by adjusting the voltages on SLM which createsproperly varying index of refraction resulting in twisting of the beamwith a specific topological charge. Different topological charges can becreated and mixed together and de-muxed to separate charges.

As Spiral Phase plates can transform a plane wave (l=0) to a twistedwave of a specific helicity (i.e. l=+1), Quarter Wave Plates (QWP) cantransform a linear polarization (s=0) to circular polarization (i.e.s=+1).

Cross talk and multipath interference can be reduced usingMultiple-Input-Multiple-Output (MIMO).

Most of the channel impairments can be detected using a control or pilotchannel and be corrected using algorithmic techniques (closed loopcontrol system).

Multiplexing of the topological charge to the RF as well as free spaceoptics in real time provides redundancy and better capacity. Whenchannel impairments from atmospheric disturbances or scintillationimpact the information signals, it is possible to toggle between freespace optics to RF and back in real time. This approach still usestwisted waves on both the free space optics as well as the RF signal.Most of the channel impairments can be detected using a control or pilotchannel and be corrected using algorithmic techniques (closed loopcontrol system) or by toggling between the RF and free space optics.

In a further embodiment illustrated in FIG. 52, both RF signals and freespace optics may be implemented within a dual RF and free space opticsmechanism 5202. The dual RF and free space optics mechanism 5202 includea free space optics projection portion 5204 that transmits a light wavehaving an orbital angular momentum applied thereto with multileveloverlay modulation and a RF portion 5206 including circuitry necessaryfor transmitting information with orbital angular momentum andmultilayer overlay on an RF signal 5210. The dual RF and free spaceoptics mechanism 5202 may be multiplexed in real time between the freespace optics signal 5208 and the RF signal 5210 depending upon operatingconditions. In some situations, the free space optics signal 5208 wouldbe most appropriate for transmitting the data. In other situations, thefree space optics signal 5208 would not be available and the RF signal5210 would be most appropriate for transmitting data. The dual RF andfree space optics mechanism 5202 may multiplex in real time betweenthese two signals based upon the available operating conditions.

Multiplexing of the topological charge to the RF as well as free spaceoptics in real time provides redundancy and better capacity. Whenchannel impairments from atmospheric disturbances or scintillationimpact the information signals, it is possible to toggle between freespace optics to RF and back in real time. This approach still usestwisted waves on both the free space optics as well as the RF signal.Most of the channel impairments can be detected using a control or pilotchannel and be corrected using algorithmic techniques (closed loopcontrol system) or by toggling between the RF and free space optics.

Quantum Key Distribution

Referring now to FIG. 53, there is illustrated a further improvement ofa system utilizing orbital angular momentum processing. In theillustration of FIG. 53, a transmitter 5302 and receiver 5304 areinterconnected over an optical link 5306. The optical link 5306 maycomprise a fiber-optic link or a free-space optic link as describedherein above. The transmitter receives a data stream 5308 that isprocessed via orbital angular momentum processing circuitry 5310. Theorbital angular momentum processing circuitry 5310 provide orbitalangular momentum twist to various signals on separate channels asdescribed herein above. In some embodiments, the orbital angularmomentum processing circuitry may further provide multi-layer overlaymodulation to the signal channels in order to further increase systembandwidth.

The OAM processed signals are provided to quantum key distributionprocessing circuitry 5312. The quantum key distribution processingcircuitry 5312 utilizes the principals of quantum key distribution aswill be more fully described herein below to enable encryption of thesignal being transmitted over the optical link 5306 to the receiver5304. The received signals are processed within the receiver 5304 usingthe quantum key distribution processing circuitry 5314. The quantum keydistribution processing circuitry 5314 decrypts the received signalsusing the quantum key distribution processing as will be more fullydescribed herein below. The decrypted signals are provided to orbitalangular momentum processing circuitry 5316 which removes any orbitalangular momentum twist from the signals to generate the plurality ofoutput signals 5318. As mentioned previously, the orbital angularmomentum processing circuitry 5316 may also demodulate the signals usingmultilayer overlay modulation included within the received signals.

Orbital angular momentum in combination with optical polarization isexploited within the circuit of FIG. 53 in order to encode informationin rotation invariant photonic states, so as to guarantee fullindependence of the communication from the local reference frames of thetransmitting unit 5302 and the receiving unit 5304. There are variousways to implement quantum key distribution (QKD), a protocol thatexploits the features of quantum mechanics to guarantee unconditionalsecurity in cryptographic communications with error rate performancesthat are fully compatible with real world application environments.

Encrypted communication requires the exchange of keys in a protectedmanner. This key exchanged is often done through a trusted authority.Quantum key distribution is an alternative solution to the keyestablishment problem. In contrast to, for example, public keycryptography, quantum key distribution has been proven to beunconditionally secure, i.e., secure against any attack, even in thefuture, irrespective of the computing power or in any other resourcesthat may be used. Quantum key distribution security relies on the lawsof quantum mechanics, and more specifically on the fact that it isimpossible to gain information about non-orthogonal quantum stateswithout perturbing these states. This property can be used to establishrandom keys between a transmitter and receiver, and guarantee that thekey is perfectly secret from any third party eavesdropping on the line.

In parallel to the “full quantum proofs” mentioned above, the securityof QKD systems has been put on stable information theoretic footing,thanks to the work on secret key agreements done in the framework ofinformation theoretic cryptography and to its extensions, triggered bythe new possibilities offered by quantum information. Referring now toFIG. 54, within a basic QKD system, a QKD link 5402 is a point to pointconnection between a transmitter 5404 and a receiver 5406 that want toshare secret keys. The QKD link 5402 is constituted by the combinationof a quantum channel 5408 and a classic channel 5410. The transmitter5404 generates a random stream of classical bits and encodes them into asequence of non-orthogonal states of light that are transmitted over thequantum channel 5408. Upon reception of these quantum states, thereceiver 5406 performs some appropriate measurements leading thereceiver to share some classical data over the classical link 5410correlated with the transmitter bit stream. The classical channel 5410is used to test these correlations.

If the correlations are high enough, this statistically implies that nosignificant eavesdropping has occurred on the quantum channel 5408 andthus, that has a very high probability, a perfectly secure, symmetrickey can be distilled from the correlated data shared by the transmitter5404 and the receiver 5406. In the opposite case, the key generationprocess has to be aborted and started again. The quantum keydistribution is a symmetric key distribution technique. Quantum keydistribution requires, for authentication purposes, that the transmitter5404 and receiver 5406 share in advance a short key whose length scalesonly logarithmically in the length of the secret key generated by an OKDsession.

Quantum key distribution on a regional scale has already beendemonstrated in a number of countries. However, free-space optical linksare required for long distance communication among areas which are notsuitable for fiber installation or for moving terminals, including theimportant case of satellite based links. The present approach exploitsspatial transverse modes of the optical beam, in particular of the OAMdegree of freedom, in order to acquire a significant technical advantagethat is the insensitivity of the communication to relevant alignment ofthe user's reference frames. This advantage may be very relevant forquantum key distribution implementation to be upgraded from the regionalscale to a national or continental one, or for links crossing hostileground, and even for envisioning a quantum key distribution on a globalscale by exploiting orbiting terminals on a network of satellites.

The OAM Eigen modes are characterized by a twisted wavefront composed of“l” intertwined helices, where “l” is an integer, and by photonscarrying “±lh” of (orbital) angular momentum, in addition to the moreusual spin angular momentum (SAM) associated with polarization. Thepotentially unlimited value of“l” opens the possibility to exploit OAMalso for increasing the capacity of communication systems (although atthe expense of increasing also the channel cross-section size), andterabit classical data transmission based on OAM multiplexing can bedemonstrated both in free-space and optical fibers. Such a feature canalso be exploited in the quantum domain, for example to expand thenumber of qubits per photon, or to achieve new functions, such as therotational invariance of the qubits.

In a free-space QKD, two users (Alice and Bob) must establish a sharedreference frame (SRF) in order to communicate with good fidelity. Indeedthe lack of a SRF is equivalent to an unknown relative rotation whichintroduces noise into the quantum channel, disrupting the communication.When the information is encoded in photon polarization, such a referenceframe can be defined by the orientations of Alice's and Bob's“horizontal” linear polarization directions. The alignment of thesedirections needs extra resources and can impose serious obstacles inlong distance free space QKD and/or when the misalignment varies intime. As indicated, we can solve this by using rotation invariantstates, which remove altogether the need for establishing a SRF. Suchstates are obtained as a particular combination of OAM and polarizationmodes (hybrid states), for which the transformation induced by themisalignment on polarization is exactly balanced by the effect of thesame misalignment on spatial modes. These states exhibit a globalsymmetry under rotations of the beam around its axis and can bevisualized as space-variant polarization states, generalizing thewell-known azimuthal and radial vector beams, and forming atwo-dimensional Hilbert space. Moreover, this rotation-invariant hybridspace can be also regarded as a decoherence-free subspace of thefour-dimensional OAM-polarization product Hilbert space, insensitive tothe noise associated with random rotations.

The hybrid, states can be generated by a particular space-variantbirefringent plate having topological charge “q” at its center, named“q-plate”. In particular, a polarized Gaussian beam (having zero OAM)passing through a q-plate with q=½ will undergo the followingtransformation:

(α|R

+β|R

)_(π)

|0

_(O) →α|L

_(π)

|r

_(O) +β|R

_(π)

|L

_(O)

|L>_(π—) and |R>_(π) denote the left and right circular polarizationstates (eigenstates of SAM with eigenvalues “±h”), |0>_(O) representsthe transverse Gaussian mode with zero OAM and the |L>_(O—) and |R>_(O)eigenstates of OAM with |l|=1 and with eigenvalues “±lh”). The statesappearing on the right hand side of equation are rotation-invariantstates. The reverse operation to this can be realized by a secondq-plate with the same q. In practice, the q-plate operates as aninterface between the polarization space and the hybrid one, convertingqubits from one space to the other and vice versa in a universal (qubitinvariant) way. This in turn means that the initial encoding and finaldecoding of information in our QKD implementation protocol can beconveniently performed in the polarization space, while the transmissionis done in the rotation-invariant hybrid space.

OAM is a conserved quantity for light propagation in vacuum, which isobviously important for communication applications. However, OAM is alsohighly sensitive to atmospheric turbulence, a feature which limits itspotential usefulness in many practical cases unless new techniques aredeveloped to deal with such issues.

Quantum cryptography describes the use of quantum mechanical effects (inparticular quantum communication and quantum computation) to performcryptographic tasks or to break cryptographic systems. Well-knownexamples of quantum cryptography are the use of quantum communication toexchange a key securely (quantum key distribution) and the hypotheticaluse of quantum computers that would allow the breaking of variouspopular public-key encryption and signature schemes (e.g., RSA).

The advantage of quantum cryptography lies in the fact that it allowsthe completion of various cryptographic tasks that are proven to beimpossible using only classical (i.e. non-quantum) communication. Forexample, quantum mechanics guarantees that measuring quantum datadisturbs that data; this can be used to detect eavesdropping in quantumkey distribution.

Quantum key distribution (QKD) uses quantum mechanics to guaranteesecure communication. It enables two parties to produce a shared randomsecret key known only to them, which can then be used to encrypt anddecrypt messages.

An important and unique property of quantum distribution is the abilityof the two communicating users to detect the presence of any third partytrying to gain knowledge of the key. This results from a fundamentalaspect of quantum mechanics: the process of measuring a quantum systemin general disturbs the system. A third party trying to eavesdrop on thekey must in some way measure it, thus introducing detectable anomalies.By using quantum superposition or quantum entanglement and transmittinginformation in quantum states, a communication system can be implementedwhich detects eavesdropping. If the level of eavesdropping is below acertain threshold, a key can be produced that is guaranteed to be secure(i.e. the eavesdropper has no information about it), otherwise no securekey is possible and communication is aborted.

The security of quantum key distribution relies on the foundations ofquantum mechanics, in contrast to traditional key distribution protocolwhich relies on the computational difficulty of certain mathematicalfunctions, and cannot provide any indication of eavesdropping orguarantee of key security.

Quantum key distribution is only used to reduce and distribute a key,not to transmit any message data. This key can then be used with anychosen encryption algorithm to encrypt (and decrypt) a message, which istransmitted over a standard communications channel. The algorithm mostcommonly associated with QKD is the one-time pad, as it is provablysecure when used with a secret, random key.

Quantum communication involves encoding information in quantum states,or qubits, as opposed to classical communication's use of bits. Usually,photons are used for these quantum states and thus is applicable withinoptical communication systems. Quantum key distribution exploits certainproperties of these quantum states to ensure its security. There areseveral approaches to quantum key distribution, but they can be dividedinto two main categories, depending on which property they exploit. Thefirst of these are prepare and measure protocol. In contrast toclassical physics, the act of measurement is an integral part of quantummechanics. In general, measuring an unknown quantum state changes thatstate in some way. This is known as quantum indeterminacy, and underliesresults such as the Heisenberg uncertainty principle, informationdistribution theorem, and no cloning theorem. This can be exploited inorder to detect any eavesdropping on communication (which necessarilyinvolves measurement) and, more importantly, to calculate the amount ofinformation that has been intercepted. Thus, by detecting the changewithin the signal, the amount of eavesdropping or information that hasbeen intercepted may be determined by the receiving party.

The second category involves the use of entanglement based protocols.The quantum states of two or more separate objects can become linkedtogether in such a way that they must be described by a combined quantumstate, not as individual objects. This is known as entanglement, andmeans that, for example, performing a measurement on one object affectsthe other object. If an entanglement pair of objects is shared betweentwo parties, anyone intercepting either object alters the overallsystem, revealing the presence of a third party (and the amount ofinformation that they have gained). Thus, again, undesired reception ofinformation may be determined by change in the entangled pair of objectsthat is shared between the parties when intercepted by an unauthorizedthird party.

One example of a quantum key distribution (QKD) protocol is the BB84protocol. The BB84 protocol was originally described using photonpolarization states to transmit information. However, any two pairs ofconjugate states can be used for the protocol, and optical fiber-basedimplementations described as BB84 can use phase-encoded states. Thetransmitter (traditionally referred to as Alice) and the receiver(traditionally referred to as Bob) are connected by a quantumcommunication channel which allows quantum states to be transmitted. Inthe case of photons, this channel is generally either an optical fiber,or simply free-space, as described previously with respect to FIG. 53.In addition, the transmitter and receiver communicate via a publicclassical channel, for example using broadcast radio or the Internet.Neither of these channels needs to be secure. The protocol is designedwith the assumption that an eavesdropper (referred to as Eve) caninterfere in any way with both the transmitter and receiver.

Referring now to FIG. 55, the security of the protocol comes fromencoding the information in non-orthogonal states. Quantum indeterminacymeans that these states cannot generally be measured without disturbingthe original state. BB84 uses two pair of states 5502, each pairconjugate to the other pair to form a conjugate pair 5504. The twostates 5502 within a pair 5504 are orthogonal to each other. Pairs oforthogonal states are referred to as a basis. The usual polarizationstate pairs used are either the rectilinear basis of vertical (0degrees) and horizontal (90 degrees), the diagonal basis of 45 degreesand 135 degrees, or the circular basis of left handedness and/or righthandedness. Any two of these basis are conjugate to each other, and soany two can be used in the protocol. In the example of FIG. 56,rectilinear basis are used at 5602 and 5604, respectively, and diagonalbasis are used at 5606 and 5608.

The first step in BB84 protocol is quantum transmission. Referring nowto FIG. 57 wherein there is illustrated a flow diagram describing theprocess, wherein the transmitter creates a random bit (0 or 1) at step5702, and randomly selects at 5704 one of the two basis, eitherrectilinear or diagonal, to transmit the random bit. The transmitterprepares at step 5706 a photon polarization state depending both on thebit value and the selected basis, as shown in FIG. 55. So, for example,a 0 is encoded in the rectilinear basis (+) as a vertical polarizationstate and a 1 is encoded in a diagonal basis (X) as a 135 degree state.The transmitter transmits at step 5708 a single proton in the statespecified to the receiver using the quantum channel. This process isrepeated from the random bit stage at step 5702 with the transmitterrecording the state, basis, and time of each photon that is sent overthe optical link.

According to quantum mechanics, no possible measurement distinguishesbetween the four different polarization states 5602 through 5608 of FIG.56, as they are not all orthogonal. The only possible measurement isbetween any two orthogonal states (and orthonormal basis). So, forexample, measuring in the rectilinear basis gives a result of horizontalor vertical. If the photo was created as horizontal or vertical (as arectilinear eigenstate), then this measures the correct state, but if itwas created as 45 degrees or 135 degrees (diagonal eigenstate), therectilinear measurement instead returns either horizontal or vertical atrandom. Furthermore, after this measurement, the proton is polarized inthe state it was measured in (horizontal or vertical), with all of theinformation about its initial polarization lost.

Referring now to FIG. 58, as the receiver does not know the basis thephotons were encoded in, the receiver can only select a basis at randomto measure in, either rectilinear or diagonal. At step 5802, thetransmitter does this for each received photon, recording the timemeasurement basis used and measurement result at step 5804. At step5806, a determination is made if there are further protons present and,if so, control passes back to step 5802. Once inquiry step 5806determines the receiver had measured all of the protons, the transceivercommunicates at step 5808 with the transmitter over the publiccommunications channel. The transmitter broadcast the basis for eachphoton that was sent at step 5810 and the receiver broadcasts the basiseach photon was measured in at step 5812. Each of the transmitter andreceiver discard photon measurements where the receiver used a differentbasis at step 5814 which, on average, is one-half, leaving half of thebits as a shared key, at step 5816. This process is more fullyillustrated in FIG. 59.

The transmitter transmits the random bit 01101001. For each of thesebits respectively, the transmitter selects the sending basis ofrectilinear, rectilinear, diagonal, rectilinear, diagonal, diagonal,diagonal, and rectilinear. Thus, based upon the associated random bitsselected and the random sending basis associated with the signal, thepolarization indicated in line 5802 is provided. Upon receiving thephoton, the receiver selects the random measuring basis as indicated inline 5904. The photon polarization measurements from these basis willthen be as indicated in line 5906. A public discussion of thetransmitted basis and the measurement basis are discussed at 5908 andthe secret key is determined to be 0101 at 5910 based upon the matchingbases for transmitted photons 1, 3, 6, and 8.

Referring now to FIG. 60, there is illustrated the process fordetermining whether to keep or abort the determined key based uponerrors detected within the determined bit string. To check for thepresence of eavesdropping, the transmitter and receiver compare acertain subset of their remaining bit strings at step 6002. If a thirdparty has gained any information about the photon's polarization, thisintroduces errors within the receiver's measurements. If more than Pbits differ at inquiry step 6004, the key is aborted at step 6006, andthe transmitter and receiver try again, possibly with a differentquantum channel, as the security of the key cannot be guaranteed. P ischosen so that if the number of bits that is known to the eavesdropperis less than this, privacy amplification can be used to reduce theeavesdropper's knowledge of the key to an arbitrarily small amount byreducing the length of the key. If inquiry step 6004 determines that thenumber of bits is not greater than P, then the key may be used at step6008.

The E91 protocol comprises another quantum key distribution scheme thatuses entangled pairs of protons. The entangled pairs can be created bythe transmitter, by the receiver, or by some other source separate fromboth of the transmitter and receiver, including an eavesdropper. Thephotons are distributed so that the transmitter and receiver each end upwith one photon from each pair. The scheme relies on two properties ofentanglement. First, the entangled states are perfectly correlated inthe sense that if the transmitter and receiver both measure whethertheir particles have vertical or horizontal polarizations, they alwaysget the same answer with 100 percent probability. The same is true ifthey both measure any other pair of complementary (orthogonal)polarizations. However, the particular results are not completelyrandom. It is impossible for the transmitter to predict if thetransmitter, and thus the receiver, will get vertical polarizations orhorizontal polarizations. Second, any attempt at eavesdropping by athird party destroys these correlations in a way that the transmitterand receiver can detect. The original Ekert protocol (E91) consists ofthree possible states and testing Bell inequality violation fordetecting eavesdropping.

Presently, the highest bit rate systems currently using quantum keydistribution demonstrate the exchange of secure keys at 1 Megabit persecond over a 20 kilometer optical fiber and 10 Kilobits per second overa 100 kilometer fiber.

The longest distance over which quantum key distribution has beendemonstrated using optical fiber is 148 kilometers. The distance is longenough for almost all of the spans found in today's fiber-opticnetworks. The distance record for free-space quantum key distribution is144 kilometers using BB84 enhanced with decoy states.

Referring now to FIG. 61, there is illustrated a functional blockdiagram of a transmitter 6102 and receiver 6104 that can implementalignment of free-space quantum key distribution. The system canimplement the BB84 protocol with decoy states. The controller 6106enables the bits to be encoded in two mutually unbiased basesZ={|0>,|1>} and X={|+>, |−>}, where |0> and |1> are two orthogonalstates spanning the qubit space and |±

=1√2 (|0)±|1

). The transmitter controller 6106 randomly chooses between the Z and Xbasis to send the classical bits 0 and 1. Within hybrid encoding, the Zbasis corresponds to {|l

_(π)

|r

_(O), |R

_(π)

|l

_(O)} while the X basis states correspond to 1/√2(|L

_(π)

|r

_(O)=|R

_(π)

|l

_(O)). The transmitter 6102 uses four different polarized attenuatedlasers 6108 to generate quantum bits through the quantum bit generator6110. Photons from the quantum bit generator 4610 are delivered via asingle mode fiber 6112 to a telescope 6114. Polarization states |H>,|V>, |R>, |L> are transformed into rotation invariant hybrid states bymeans of a q-plate 6116 with q=½. The photons can then be transmitted tothe receiving station 6104 where a second q-plate transform 6118transforms the signals back into the original polarization states |H>,|V>, |R>, |L>, as defined by the receiver reference frame. Qubits canthen be analyzed by polarizers 6120 and single photon detectors 6122.The information from the polarizers 6120 and photo detectors 6122 maythen be provided to the receiver controller 6124 such that the shiftedkeys can be obtained by keeping only the bits corresponding to the samebasis on the transmitter and receiver side as determined bycommunications over a classic channel between the transceivers 6126,6128 in the transmitter 6102 and receiver 6104.

Referring now to FIG. 62, there is illustrated a network cloud basedquantum key distribution system including a central server 6202 andvarious attached nodes 6204 in a hub and spoke configuration. Trends innetworking are presenting new security concerns that are challenging tomeet with conventional cryptography, owing to constrained computationalresources or the difficulty of providing suitable key management. Inprinciple, quantum cryptography, with its forward security andlightweight computational footprint, could meet these challenges,provided it could evolve from the current point to point architecture toa form compatible with multimode network architecture. Trusted quantumkey distribution networks based on a mesh of point to point links lacksscalability, require dedicated optical fibers, are expensive and notamenable to mass production since they only provide one of thecryptographic functions, namely key distribution needed for securecommunications. Thus, they have limited practical interest.

A new, scalable approach such as that illustrated in FIG. 62 providesquantum information assurance that is network based quantumcommunications which can solve new network security challenges. In thisapproach, a BB84 type quantum communication between each of N clientnodes 6204 and a central sever 6202 at the physical layer support aquantum key management layer, which in turn enables secure communicationfunctions (confidentiality, authentication, and nonrepudiation) at theapplication layer between approximately N2 client pairs. This networkbased communication “hub and spoke” topology can be implemented in anetwork setting, and permits a hierarchical trust architecture thatallows the server 6202 to act as a trusted authority in cryptographicprotocols for quantum authenticated key establishment. This avoids thepoor scaling of previous approaches that required a pre-existing trustrelationship between every pair of nodes. By making a server 6202, asingle multiplex QC (quantum communications) receiver and the clientnodes 6204 QC transmitters, this network can simplify complexity acrossmultiple network nodes. In this way, the network based quantum keydistribution architecture is scalable in terms of both quantum physicalresources and trust. One can at time multiplex the server 6202 withthree transmitters 6204 over a single mode fiber, larger number ofclients could be accommodated with a combination of temporal andwavelength multiplexing as well as orbital angular momentum multiplexedwith wave division multiplexing to support much higher clients.

Referring now to FIGS. 63 and 64, there are illustrated variouscomponents of multi-user orbital angular momentum based quantum keydistribution multi-access network. FIG. 63 illustrates a high speedsingle photon detector 6302 positioned at a network node that can beshared between multiple users 6304 using conventional networkarchitectures, thereby significantly reducing the hardware requirementsfor each user added to the network. In an embodiment, the single photondetector 6302 may share up to 64 users. This shared receiverarchitecture removes one of the main obstacles restricting thewidespread application of quantum key distribution. The embodimentpresents a viable method for realizing multi-user quantum keydistribution networks with resource efficiency.

Referring now also to FIG. 64, in a nodal quantum key distributionnetwork, multiple trusted repeaters 6402 are connected via point topoint links 6404 between node 6406. The repeaters are connected viapoint to point links between a quantum transmitter and a quantumreceiver. These point to point links 6404 can be realized using longdistance optical fiber lengths and may even utilize ground to satellitequantum key distribution communication. While point to point connections6404 are suitable to form a backbone quantum core network, they are lesssuitable to provide the last-mile service needed to give a multitude ofusers access to the quantum key distribution infrastructure.Reconfigurable optical networks based on optical switches or wavelengthdivision multiplexing may achieve more flexible network structures,however, they also require the installation of a full quantum keydistribution system per user which is prohibitively expensive for manyapplications.

The quantum key signals used in quantum key distribution need onlytravel in one direction along a fiber to establish a secure key betweenthe transmitter and the receiver. Single photon quantum key distributionwith the sender positioned at the network node 6406 and the receiver atthe user premises therefore lends itself to a passive multi-user networkapproach. However, this downstream implementation has two majorshortcomings Firstly, every user in the network requires a single photondetector, which is often expensive and difficult to operate.Additionally, it is not possible to deterministically address a user.All detectors, therefore, have to operate at the same speed as atransmitter in order not to miss photons, which means that most of thedetector bandwidth is unused.

Most systems associated with a downstream implementation can beovercome. The most valuable resource should be shared by all users andshould operate at full capacity. One can build an upstream quantumaccess network in which the transmitters are placed at the end userlocation and a common receiver is placed at the network node. This way,an operation with up to 64 users is feasible, which can be done withmulti-user quantum key distribution over a 1×64 passive opticalsplitter.

Thus, using various configurations of the above described orbitalangular momentum processing, multi-layer overlay modulation, and quantumkey distribution within various types of communication networks and moreparticularly optical fiber networks and free-space optic communicationnetwork, a variety of benefits and improvements in system bandwidth andcapacity maybe achieved.

It will be appreciated by those skilled in the art having the benefit ofthis disclosure that this system and method for communication usingorbital angular momentum with multiple layer overlay modulation providesimproved bandwidth and data transmission capability. It should beunderstood that the drawings and detailed description herein are to beregarded in an illustrative rather than a restrictive manner, and arenot intended to be limiting to the particular forms and examplesdisclosed. On the contrary, included are any further modifications,changes, rearrangements, substitutions, alternatives, design choices,and embodiments apparent to those of ordinary skill in the art, withoutdeparting from the spirit and scope hereof, as defined by the followingclaims. Thus, it is intended that the following claims be interpreted toembrace all such further modifications, changes, rearrangements,substitutions, alternatives, design choices, and embodiments.

1. A system, comprising: first signal processing circuitry fortransmitting a signal including a plurality of data streams over a link,the first signal processing circuitry further comprising: firstcircuitry for processing each of the plurality of input data streams togenerate a plurality of parallel pairs of data streams including anin-phase stream (I) and a quadrature-phase stream (Q) for each of theplurality of input data streams, modulating each of the plurality ofparallel pairs of data streams with a selected one of at least threemutually orthogonal functions and each of the plurality of parallelpairs at a different signal width, respectively, to generate a pluralityof data signals, each associated with one of the plurality of parallelpairs of data streams, and generating a plurality of composite datastreams by overlaying at least one first data signal of the plurality ofdata signals in a first data layer with at least one second data signalof the plurality of data signals in a second data layer; and secondcircuitry for processing the plurality of composite data streams toassociate with each of the plurality of composite data streams a Hermitepolynomial to cause each of the plurality of composite data streams tobe mutually orthogonal to each other on the link to enable transmissionof each of the plurality of composite data streams on the link at a sametime.
 2. The system of claim 1, wherein the link further comprises afiber optic link.
 3. The system of claim 2, wherein the fiber optic linkcomprise a multi-mode fiber.
 4. The system of claim 1, wherein the linkfurther comprises a free space optics link.
 5. The system of claim 1,wherein the link further comprises an RF link.
 6. The system of claim 1,further including third circuitry for selecting an encryption key fortransmissions over the link using quantum key distribution.
 7. Thesystem of claim 1, wherein the at least three mutually orthogonalfunctions comprises a plurality of time-limited and band-limitedfunctions.
 8. The communications system of claim 1, wherein the at leastthree mutually orthogonal functions comprises at least one of aplurality of modified Hermite polynomials, Jacobi polynomials,Gegenbauer polynomials, Legendre polynomials, Chebyshev polynomials andLaguerre functions.
 9. The system of claim 1, wherein the at least threemutually orthogonal functions comprises a plurality of rectangular,cylindrical, and spherical functions.
 10. The system of claim 1, furtherincluding second signal processing circuitry for receiving the pluralityof composite data streams on the link, the second signal processingcircuitry further comprising: a signal separator for separating each ofthe plurality of composite data streams having the different Hermitepolynomial applied thereto by the second circuitry from each other;third circuitry for removing the different Hermite polynomial appliedthereto by the second circuitry from each of the plurality of compositedata streams; fourth circuitry for demodulating the plurality ofcomposite data streams into the plurality of input data streams.
 11. Asystem, comprising: first signal processing circuitry for transmitting asignal including a plurality of data streams over a link, the firstsignal processing circuitry further comprising: first circuitry forprocessing each of the plurality of input data streams to generate aplurality of parallel pairs of data streams including an in-phase stream(I) and a quadrature-phase stream (Q) for each of the plurality of inputdata streams, modulating each of the plurality of parallel pairs of datastreams with a selected one of at least three mutually orthogonalfunctions and each of the plurality of parallel pairs at a differentsignal width, respectively, to generate a plurality of data signals,each associated with one of the plurality of parallel pairs of datastreams, and generating a plurality of composite data streams byoverlaying at least one first data signal of the plurality of datasignals in a first data layer with at least one second data signal ofthe plurality of data signals in a second data layer; and secondcircuitry for processing the plurality of composite data streams toassociate with each of the plurality of composite data streams a Hermitepolynomial to cause each of the plurality of composite data streams tobe mutually orthogonal to each other on the link to enable transmissionof each of the plurality of composite data streams on the link at a sametime; second signal processing circuitry for receiving the signal overthe link, the second signal processing circuitry further comprising:third circuitry to extract the plurality of composite data streamshaving the different Hermite polynomial applied thereto by the secondsignal processing circuitry associated therewith from link; and fourthcircuitry for demodulating the plurality of composite data streams intothe plurality of input data streams.
 12. The system of claim 11, whereinthe link further comprises at least one of a fiber optic link and RFlink.
 13. The system of claim 12, wherein the fiber optic link comprisea multi-mode fiber.
 14. The system of claim 11, wherein the link furthercomprises a free space optics link.
 15. The system of claim 11, whereinthe first signal processing circuitry further includes fifth circuitryfor selecting an encryption key for transmissions over the link usingquantum key distribution.
 16. The system of claim 11, wherein the atleast three mutually orthogonal functions comprises at least one of aplurality of time-limited and band-limited functions, rectangularfunctions, cylindrical functions and spherical functions.
 17. The systemof claim 11, wherein the at least three mutually orthogonal functionscomprises at least one of a plurality of modified Hermite polynomials,Jacobi polynomials, Gegenbauer polynomials, Legendre polynomials,Chebyshev polynomials and Laguerre functions.
 18. A method for providinga plurality of input streams from first signal processing circuitry tosecond signal processing circuitry over a link, comprising: receivingthe plurality of input streams; processing each of the plurality ofinput data streams to generate a plurality of parallel pairs of datastreams including an in-phase stream (I) and a quadrature-phase stream(Q) for each of the plurality of input data streams; modulating each ofthe plurality of parallel pairs of data streams with a selected one ofat least three mutually orthogonal functions and each of the pluralityof parallel pairs at a different signal width, respectively, to generatea plurality of data signals, each associated with one of the pluralityof parallel pairs of data streams, and generating a plurality ofcomposite data streams by overlaying at least one first data signal ofthe plurality of data signals in a first data layer with at least onesecond data signal of the plurality of data signals in a second datalayer; applying Hermite polynomials to each of the plurality ofcomposite data streams to cause each of the plurality of composite datastreams to be mutually orthogonal to each other; and placing theplurality of composite data streams on the link, each of the pluralityof composite data streams having the different Hermite polynomialassociated therewith to enable each of the plurality of composite datastreams on the link at a same time.
 19. The method of claim 18, whereinthe link further comprises at least one of a fiber optic link and RFlink.
 20. The method of claim 19, wherein the fiber optic link comprisea multi-mode fiber.
 21. The method of claim 18, wherein the link furthercomprises a free space optics link.
 22. The method of claim 18, furtherincluding selecting an encryption key for data passed over the linkusing quantum key distribution.
 23. The method of claim 18, wherein theat least three mutually orthogonal functions comprises at least one of aplurality of time-limited and band-limited functions, rectangularfunctions, cylindrical functions and spherical functions.
 24. The methodof claim 18, wherein the at least three mutually orthogonal functionscomprises a plurality of at least one of modified Hermite polynomials,Jacobi polynomials, Gegenbauer polynomials, Legendre polynomials,Chebyshev polynomials and Laguerre functions.
 25. The method of claim18, further including: receiving the plurality of composite data streamson the link; separating each of the plurality of composite data streamshaving the different Hermite polynomial from each other; demodulatingthe Hermite polynomial from each of the plurality of composite datastreams; and demodulating the plurality of composite data streams intothe plurality of input data streams.
 26. A system, comprising: firstsignal processing circuitry for transmitting a signal including aplurality of data streams over a link, the first signal processingcircuitry further comprising: first circuitry for processing each of theplurality of input data streams to generate a plurality of parallelpairs of data streams including an in-phase stream (I) and aquadrature-phase stream (Q) for each of the plurality of input datastreams, modulating each of the plurality of parallel pairs of datastreams with a different, selected one of at least three mutuallyorthogonal functions and each of the plurality of parallel pairs at adifferent signal width, respectively, to generate a plurality of datasignals, each associated with one of the plurality of parallel pairs ofdata streams, and generating a plurality of composite data streams byoverlaying at least one first data signal of the plurality of datasignals in a first data layer with at least one second data signal ofthe plurality of data signals in a second data layer; and secondcircuitry for processing the plurality of composite data streams toassociate with each of the plurality of composite data streams anorthogonal function to cause each of the plurality of composite datastreams to be mutually orthogonal to each other on the link to enabletransmission of each of the plurality of composite data streams on thelink at a same time.
 27. The system of claim 26, further including thirdcircuitry for selecting an encryption key for transmissions over thelink using quantum key distribution.
 28. The system of claim 26, whereinthe at least three mutually orthogonal functions comprises at least oneof a plurality of time-limited and band-limited functions, rectangularfunctions, cylindrical functions and spherical functions.
 29. The systemof claim 26, wherein the at least three mutually orthogonal functionscomprises at least one of a plurality of modified Hermite polynomials,Jacobi polynomials, Gegenbauer polynomials, Legendre polynomials,Chebyshev polynomials and Laguerre functions.
 30. The system of claim26, further including a second signal processing circuitry for receivingthe plurality of composite data streams on the at least one of the link,the second signal processing circuitry for receiving further comprising:a signal separator for separating each of the plurality of compositedata streams having the different orthogonal function from the receivedsignal on the link; third circuitry for removing the differentorthogonal function from each of the plurality of composite datastreams; fourth circuitry for demodulating the plurality of compositedata streams into the plurality of input data streams.